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Geometry Unit 4 Test Answer Key

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Geometry Unit 4 Test Answers

Construction

A drawing using only a compass and straightedge.

Straightedge

A ruler or tool used to draw straight lines.

Compass

A geometric tool used to draw circles and parts of circles called arcs.

Protractor

A tool used to measure angles.

Equilateral

Having all sides equal.

Radius

The distance from the center of a circle to any point on the circle.

Equidistant

Equally distant from two points.

Isosceles

Having two equal sides.

Arc

Part of a circle connecting two points on the circle.

Golden Rule of Constructions

Always leave your construction marks.

Parallel

Two lines in the same plane that do not intersect.

Quadrilateral

A four-sided polygon.

Acute Angle

An angle that measures less than 90 degrees.

Obtuse Angle

An angle that measures more than 90 degrees but less than 180 degrees.

Right Angle

An angle that measures 90 degrees.

Congruent

Having the same size and shape; equal.

Perpendicular

Intersecting at or forming right angles.

Hypotenuse

The longest side of a right triangle; opposite the right angle.

Bisector

A point, line or line segment that divides a segment or angle into two equal parts.

Perpendicular Bisector

A segment, ray, line, or plane that is perpendicular to a segment at its midpoint.

Endpoint

A point at either end of a line segment, or a point at one end of a ray.

Midpoint

A point that divides a segment into two congruent segments.

Altitude

Height; creates a right angle with the base.

Diameter

A straight line passing from side to side through the center of a circle.

Circumscribe

To draw a circle around.

Circumcenter

The point of concurrency of the perpendicular bisectors of a triangle

Concurrence

When two or more lines or line segments intersect.

Rotational Symmetry

A type of symmetry a figure has if it can be rotated less than 360 degrees about its center and still look like the original.

Axiom

A universal truth; an established rule

Angle Bisector

A line, segment, or ray that divides an angle into two congruent angles

Diagonal

A straight line connecting any two vertices of a polygon that are not adjacent.

Rectangle

A parallelogram with four right angles.

Ray

A part of a line, with one endpoint, that continues without end in one direction.

Inscribed Circle

A circle that is inside the triangle.

Incenter

The point of concurrency of the angle bisectors of a triangle

Square

A parallelogram with four congruent sides and four right angles.

Hexagon

6 sided polygon.

Equilateral Triangle

A triangle with three congruent sides.

Theorem

A statement that can be proven (ex: SSS, SAS, ASA, AAS, HL).

(L1) A(n) _____ is a closed plane figure formed by three or more line segments, such that each segment intersects exactly two other segments only at their endpoints, and no two segments with a common vertex are collinear.

polygon

(L1) A(n) _____ is the common endpoint of two sides of a polygon.

vertex of a polygon

(L1) A(n) _____ is a line segment that forms a polygon.

side of a polygon

(L1) A(n) _____ is a segment that joins any two nonadjacent vertices of a polygon.

diagonal of a polygon

(L1) A(n) _____ is a polygon in which all its angles are congruent.

equiangular polygon

(L1) A(n) _____ is a polygon in which all its sides are congruent.

equilateral polygon

(L1) A(n) _____ is an angle formed in the interior of a polygon.

interior angle

(L1) A(n) _____ is an angle formed in the exterior of a polygon by a side of the polygon and the extension of a consecutive side.

exterior angle

(L1) A(n) _____ polygon is a polygon that is both equiangular and equilateral.

regular

(L1) A(n) _____ polygon is a polygon in which none of its diagonals contain points in the exterior of the polygon.

convex

(L1) A(n) _____ polygon is a polygon that is equiangular but not equilateral, or equilateral but not equiangular, or neither equiangular nor equilateral.

irregular

(L1) A(n) _____ polygon is a polygon in which any of its diagonals contain points in the exterior of the polygon.

concave

(L1) The Polygon Interior Angle Sum Theorem states that the sum of the measures of the interior angles of a convex polygon with n sides is _____.

(n-2)180°

(L1) The Polygon Exterior Angle Sum Theorem states that the sum of the measures of the exterior angles, one at each vertex, of a convex polygon is _____.

360°

(L1) The _____ states that the sum of the measures of the interior angles of a triangle is 180°.

Triangle Sum Theorem

(L1) A polygon is named according to the number of _____ of the polygon.

sides

(L1) Polygons can be classified according to the measure of their __________ or the lengths of their __________ into two general categories:
1. __________ all the interior angles are congruent
2. __________ all the sides are congruent

angles;
sides;
equiangular;
equilateral

(L1) Polygons may be classified depending on their diagonals into two categories:
1. __________ if all the diagonals are in the interior of the polygon.
2. __________ if any diagonal contains points in the exterior of the polygon.

convex;
concave

(L1) How many diagonals does a triangle have?

(L1) How many diagonals does a hexagon have?

9

(L1) How many diagonals does a nonagon have?

27

(L1) How many diagonals does a heptagon have?

14

(L1) A recent tornado damaged Jack’s house and left it a little distorted. A question on the insurance form asked, “What shape is the house in?” Jack replied, “An irregular convex pentagon.”
Find the sum of the measures of the interior angles of the 5-sided house using the Polygon Interior Angle Sum Theorem.

540°

(L1) A recent tornado damaged Jack’s house and left it a little distorted. A question on the insurance form asked, “What shape is the house in?” Jack replied, “An irregular convex pentagon.”
The sum of the measures of the interior angles of the 5-sided house is 540∘. Find the value of x.

45°

(L1) A recent tornado damaged Jack’s house and left it a little distorted. A question on the insurance form asked, “What shape is the house in?” Jack replied, “An irregular convex pentagon.”
The sum of the measures of the interior angles of the 5-sided house is 540∘. Find the value of y.

54°

(L1) Your Uncle Bob is a bricklayer who was given the job of building a brick wall around a garden that is shaped like a regular triangle. However, he wasn’t sure how to make the exterior angles (∠A,∠B, and ∠C) just right. Use the Polygon Exterior Angle Sum Theorem to calculate the measures of the angles.
m∠A=
m∠B=
m∠C=

m∠A= 120
m∠B= 120
m∠C= 120

(L1) Your Uncle Bob is a bricklayer who was given the job of building a brick wall around a garden that is shaped like a regular triangle. However, he wasn’t sure how to make the exterior angles (∠A,∠B, and ∠C) just right. Use the Polygon Exterior Angle Sum Theorem to calculate the measures of the angles.
m∠1=
m∠2=
m∠3=

m∠1= 60
m∠2= 60
m∠3= 60

(L1) Apply the Triangle Sum Theorem to find the sum of the angles in the pentagon. A pentagon can be divided into three triangles.

540°

(L2) A(n) _____ is an interior angle of a polygon that is not adjacent to a particular exterior angle.

remote interior angle

(L2) A(n) _____ is a line added to a figure to aid in a proof.

auxiliary line

(L2) A(n) _____ is a polygon with three sides.

triangle

(L2) A(n) _____ is a statement whose proof can be deduced directly from a previous theorem.

corollary

(L2) A triangle in which one of its interior angles is an obtuse angle is a(n) _____ triangle.

obtuse

(L2) A triangle with at least two congruent sides is a(n) _____ triangle.

isosceles

(L2) A special kind of isosceles triangle in which all three sides are congruent is a(n) _____ triangle.

equilateral

(L2) A triangle in which all three of its interior angles are acute angles is a(n) _____ triangle.

acute

(L2) A special kind of triangle with no congruent sides is a(n) _____ triangle.

scalene

(L2) A special kind of acute triangle in which all three angles are congruent is a(n) _____ triangle.

equiangular

(L2) A triangle in which one of the angles is a right angle is a(n) _____ triangle.

right

(L2) Which theorem or corollary could be used to prove the conjecture:
m∠3=m∠1+m∠2

Exterior Angle Theorem

(L2) Which theorem or corollary could be used to prove the conjecture:
m∠1+m∠3=90°

Corollary 4.2A (acute angles of a right triangle are complementary)

(L2) Which theorem or corollary could be used to prove the conjecture:
m∠A=m∠B=m∠C=60°

Corollary 4.2B (measure of each acute angle of an equiangular triangle is 60°)

(L2) Which theorem or corollary could be used to prove the conjecture:
∠C≅∠D

Third Angles Theorem

(L2) Which theorem or corollary could be used to prove the conjecture:
m∠1≠90°

Corollary 4.2C(triangle can contain no more than one right angle or obtuse angle)

(L2) Given: ΔABC is equiangular.
Prove: m∠A=m∠B=m∠C=60°

1. given
2. equiangular triangle
3. m/A+m/B+m/C=180°
6. Division Property of Equality

(L2) Given: ΔMNO and ΔPQR;∠M≅∠P;∠N≅∠Q
Prove: ∠O≅∠R

2. Triangle Sum Theorem
3. Subtraction
6. Substitution
7. Definition of congruent angles.

(Q1) The common endpoint of two sides of a polygon is called a(n) _____.

vertex of a polygon

(Q1) Emily wants to construct a flower bed in the shape of a regular octagon (8-sided convex polygon).
Find the measure of each of the interior angles using the Polygon Interior Angle Sum Theorem.

135°

(Q1) Which theorem or corollary could be used to prove the conjecture?
Conjecture: m∠1 is not greater than or equal to 90°

Corollary 4.2C (triangle can have no more than one right angle or obtuse angle)

(Q1) The Polygon Interior Angle Sum Theorem states that the sum of the measures of the interior angles of a convex polygon with n sides is _____.

(n-2)180°

(Q1) The Polygon Exterior Angle sum Theorem states that the sum of the measures of the exterior angles, one angle at each vertex, of a convex polygon is _____.

360°

(Q1) The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of its _____ interior angles.

remote

(Q1) A statement whose proof can be deduced directly from a previous theorem.

corollary

(Q1) An interior angle of a polygon that is not adjacent to a particular exterior angle.

remote interior angle

(Q1) Use the Polygon Exterior Angle Sum Theorem to find the sum of the measures of the exterior angles of the regular pentagon.

360°

(Q1) A segment that joins any two nonadjacent vertices of a polygon is a(n) _____.

diagonal of a polygon

(Q1) A polygon in which any of its diagonals contain points in the exterior of the polygon is called a(n) _____.

concave polygon

(Q1) A polygon that is equiangular but not equilateral, or equilateral but not equiangular, or neither equiangular nor equilateral is a(n) _____.

irregular polygon

(Q1) The Third Angles Theorem states that if two angles of one triangle are congruent to two angles of a second triangle, then the third angle of the first triangle will also be congruent to the _____ angle of the second triangle.

third

(Q1) A corollary of the Triangle Sum Theorem states that a triangle can contain no more than one _____ angle or obtuse angle.

right

(Q1) A polygon that is both equiangular and equilateral is a(n) _____.

regular polygon

(Q1) A special kind of triangle with no congruent sides.

scalene

(Q1) Which theorem or corollary could be used to prove the conjecture?
Conjecture: m∠1=m∠2+m∠3

Exterior Angle Theorem

(Q1) A polygon in which none of its diagonals contain points in the exterior of the polygon is called a(n) _____.

convex polygon

(Q1) A triangle with at least two congruent sides.

isosceles

(Q1) Emily wants to construct a flower bed in the shape of a regular octagon (8-sided convex polygon).
Find the sum of the measures of the interior angles using the Polygon Interior Angle Sum Theorem.

1080°

(Q1) A corollary to the Triangle Sum Theorem states that the measure of each angle of an equilateral triangle is _____.

60°

(Q1) Angles formed in the interior of a polygon are called _____.

interior angles

(Q1) Use the Polygon Exterior Angle Sum Theorem to find the measure of each of the exterior angles of the regular pentagon.

360°

(Q1) An angle formed in the exterior of a polygon by a side of the polygon and the extension of a consecutive side is a(n) _____.

exterior angle

(Q1) A closed plane figure formed by three or more line segments, such that each segment intersects exactly two other segments only at their endpoints, and no two segments with a common vertex are collinear is a(n) _____.

polygon

(Q1) A corollary of the Triangle Sum Theorem states that the acute angles of a right triangle are _____.

complementary

(L3) Triangles in which their corresponding angles and corresponding sides are congruent are called _____.

congruent triangles

(L3) A(n) _____ is an angle formed by the intersection of two adjacent sides of a polygon.

included angle

(L3) _____ are angles that are in the same relative position on two different polygons that have the same number of angles.

Corresponding angles of polygons

(L3) Polygons in which their corresponding angles and corresponding sides are congruent are called _____.

congruent polygons

(L3) A(n) _____ is any side of a polygon that shares a side with only one angle of a pair of angles.

non-included side

(L3) Sides that are in the same relative position on two different polygons that have the same number of sides are called _____.

corresponding sides of polygons

(L3) A(n) _____ is the side shared by two consecutive angles of a polygon.

included side

(L3) Which side of ΔDEF corresponds to BC— on ΔABC?

EF—

(L3) Which side of ΔABC corresponds to DF— on ΔDEF?

AC—

(L3) Which angle of ΔDEF corresponds to ∠C on ΔABC?

∠F

(L3) If ΔDEF is isosceles, which two angles of ΔDEF appear to be the base angles?

∠E and ∠F

(L3) Given: ΔABD and ΔDCB ; AB—≅DC— and AD—≅BC—
Prove : ΔABD≅ΔDCB

1. D
2. B
3. G

(L3) Given: AC↔∥DF↔; BC—≅DE—
Prove: ΔDBE≅ΔBEC

1. B
2. G
3. F
4. D

(L3) Given: MN—∥PO—;MP—∥NO—
Prove: ΔMNP≅ΔOPN
You are given the information that (1.)__________ Since NP¯ is a transversal cutting parallel segments, MN¯ and PO¯, you know that ∠MNP≅∠OPN because of the (2.)__________ Theorem. Also, since NP¯ is a transversal cutting parallel segments, MP¯ and NO¯, you know that (3.)__________ for the same reason. You also know that NP¯≅NP¯ because of the (4.)__________Property of Congruence. Since you have two angles and an included side of ΔMNP congruent to the corresponding two angles and included side of ΔOPN, you can conclude that (5.)__________ because of ASA.

1. A
2. I
3. E
4. H
5. C

(L3) Several years ago, a popular bridge construction design was the rigid structure, V-leg design. The bridge here is supported by two triangle-shaped trusses, labeled ΔABC and ΔDEF.AB— and DE— are parallel segments cut by a transversal, AF—.∠B is congruent to ∠E and AC—≅DF—. Do you have sufficient information to be able to prove that ΔABC≅ΔDEF using the AAS Theorem?

Yes, since AB¯ and DE¯ are parallel segments cut by a transversal, AF¯, ∠A≅∠D since they are corresponding angles, and we have two congruent pairs of angles and a congruent pair of non-included sides.

(L3) Which pair of triangles demonstrates congruence with the SSS Postulate?

C

(L3) Which pair of triangles demonstrates congruence with the SAS Postulate?

B

(L3) Which pair of triangles demonstrates congruence with the ASA Postulate?

A

(L3) Which pair of triangles demonstrates congruence with the AAS Postulate?

D

(L4) The side opposite the right angle in a right triangle is called the _____.

hypotenuse

(L4) The equation for the Pythagorean Theorem is _____.

a²+b²=c² OR c²=a²+b²

(L4) If one leg of a right triangle has a measure of 7 cm and the other leg has a measure of 9 cm, what is the measure of the hypotenuse, rounded to one decimal place?

11.4 cm

(L4) If the hypotenuse of a right triangle has a measure of 8 feet and one leg has a measure of 4 feet, what is the measure of the other leg, rounded to one decimal place?

6.9 feet

(L4) The LL Theorem corresponds with the __________ Postulate.
The LA Theorem corresponds with the __________ Postulate.
The HA Theorem corresponds with the __________ Postulate.

SAS
ASA
AAS

(L4) Given: ΔABO and ΔDEO are right Δs; AB¯≅DE¯;AB¯∥DE¯.
Prove: ΔABO≅ΔDEO

2. All right angles are congruent
3. Alternate Interior Angles
4. ASA≅

(L4) Given: ΔABC and ΔDEC are right Δs ; AC=420 m;AB=200 m;CD=420 m;DE=200 m
Prove: ΔABC≅ΔDEC

3. D
4. B

(L4) Use the Pythagorean Theorem to calculate the distance from the corner of 8th St. SE and Massachusetts Ave. NE to the starting point in Lincoln Park. (Round to one decimal place.)

465.2 m

(L4) Given: ΔABC and ΔCDA are right Δs; AD¯≅CB¯
Prove: ΔABC≅ΔCDA

2. C
3. A

(Q2) _____ sides of polygons are sides that are in the same relative position on two different polygons that have the same number of sides.

Corresponding

(Q2) Any side of a polygon that shares a side with only one angle of a pair of angles is a(n) _____ side.

nonincluded

(Q2) _____ angles of polygons are angles that are in the same relative position on two different polygons that have the same number of angles.

(Q2) The _____ of a right triangle is the side opposite the right angle.

hypotenuse

(Q2) Tanya wants to buy a flat screen television that will just fit into the space in her entertainment center. The dimensions of the opening where the television would sit are width equal to 37 inches and height equal to 21 inches. Knowing that television sizes are named by the measure of the screen’s diagonal, what is the biggest size television she can get that will fit?

42 inches

(Q2) Congruent triangles are triangles in which their _____ angles and sides are congruent.

corresponding

(Q2) The _____ side is the side shared by two consecutive angles of a polygon.

included

(Q2) An angle formed by the intersection of two adjacent sides of a polygon is called a(n) _____ angle.

included

(Q2) A student pilot filed a flight plan which included flying due west from an airport in Dallas, Texas for 100 miles, then turning due north and flying 75 miles to land at an airport in Wichita Falls, Texas. How far would he then have to fly in a straight line distance to get back to Dallas?

125 miles

(Q2) Congruent polygons are polygons in which their _____ angles and sides are congruent.

corresponding

(L5) What does CPCTC stand for?

Corresponding Parts of Congruent Triangles are Congruent

(L5) You are the captain of the starship Selecsosi from the planet Yrtemoeg. While traveling in a galaxy far, far away, you spotted a previously uncharted red dwarf star at a line of sight of 45° from the ship (A). You needed to chart the new star, but you couldn’t determine its distance from the ship because the star was out of sensor range. After some quick thinking, you remembered something you learned in Geometry that helped you figure out a way to find the distance. As the ship continued on its course (AB¯), you kept watching the star until the line of sight was exactly 90° from the ship (B). The distance you traveled from point A to point B was 125 million retemoliks. From this information, you were able to calculate the distance (BC¯) to the star and chart its position.
A. First, use the Triangle Sum Theorem to calculate m∠C= __________
B. Based on that, what kind of triangle is ΔABC? __________
C. What does that tell you about the measure of BC¯? __________
D. Therefore, the distance to the star from B to C was __________

A. (F) 45°
B. (L) isosceles triangle
C. (I) BC—≅AB—
D. (E) 125 million retemoliks

(L5) Given: ΔABC;∠A≅∠B
Prove: AC¯≅BC¯

4. (F) Reflexive
5. (D) AAS
6. (G) AC¯≅BC¯

(L5) Given: ΔABC is equiangular
Prove: AB¯≅BC¯≅CA¯

(L6) Given: ΔDEF with vertices D(0,4),E(-3,0), and F(0,0); and ΔDGF with vertices D(0,4),G(3,0), and F(0,0)
Prove: ΔDEF≅ΔDGF
From the marks on the grid, we can see that the length of EF¯ is __________ units and the length of GF¯ is __________ units, so EF¯≅GF¯ .
DF¯≅DF¯ by the __________ Property of ≅.
Therefore, __________ by the Leg-Leg Congruence Theorem.

3
3
Reflexive
ΔDEF≌ΔDGF

(L6) Given: AB¯ is a line segment with endpoints A(0,6) and B(8,0) and point M(4,3).
Find the length of AM¯.
AM= √(x2-x1)2+(y2-y1)2

5

(L6) Given: AB¯ is a line segment with endpoints A(0,6) and B(8,0) and point M(4,3).
Find the length of MB¯.
MB= √(x2-x1)2+(y2-y1)2

5

(L6) Given: □ABCD is a square with vertices at A(2,8),B(8,8),C(8,2),D(2,2), and diagonal BD¯.
Prove: The area of ΔBCD=12 the area of □ABCD.
Find the length of DC_.DC=(x2-x1)2+(y2-y1)2 __________.
Find the area of □ABCD.A□ __________.
Find the area of ΔBCD. A∆=12bh= __________.
Therefore: __________.

E.
C.
D.
H.

(L6)
Samuel promised to call his wife after he was halfway between Dallas and Austin. However, the odometer on his car doesn’t work so he needs to know which city is close to the halfway point. Use the midpoint formula to calculate the coordinates of the halfway point.
M=(x1+x22,y1+y22)

(2,6)

(L6) Samuel promised to call his wife after he was halfway between Dallas and Austin. However, the odometer on his car doesn’t work so he needs to know which city is close to the halfway point. If the coordinates of the midpoint of a line segment between Dallas and Austin are (2,6),what city is closest to the halfway point?

Waco

(Q3) Which theorem, term, or corollary is represented by the picture? The bold lines in the pictures represent the hypothesis of the theorem or corollary.
The sides are bold; the angles are not bold.

Isosceles Triangle Theorem

(Q3) Given: Right ΔABD and right ΔCBD
Prove: ΔABD≅ΔCBD

A.
F.
C.
B.
G.

(Q3) Which theorem, term, or corollary is represented by the picture? The bold lines in the pictures represent the hypothesis of the theorem or corollary.
(equilateral)
The sides are bold; the angles are not bold.

Corollary to the Isosceles Triangle Theorem

(Q3) Which theorem, term, or corollary is represented by the picture? The bold lines in the pictures represent the hypothesis of the theorem or corollary.
(equilateral)
The angles are bold; the sides are not bold.

Corollary to the Converse of the Isosceles Triangle Theorem

(Q3) Given: Right ΔABC; right ∠ACB;M is the midpoint of AB¯ .
Prove: CM=12AB

E.
C.
B.

(Q3) Which theorem, term, or corollary is represented by the picture? The bold lines in the pictures represent the hypothesis of the theorem or corollary.
The angles are bold; the sides are not bold.

Converse to the Isosceles Triangle Theorem

(Q3) Which theorem, term, or corollary is represented by the picture?

CPCTC

Vertical Angles

Angles that are opposite of each other

Adjacent Angles

Angles that are next to each other (share sides and a vertex)

Complementary Angles

The sum of two angles in 90 degrees

Supplementary Angles

The sum of two angles is 180 degrees

Equilateral triangle

Triangle with congruent sides

Scalene Triangle

Triangle with no congruent sides

Isosceles Triangle

Triangle with at least two congruent sides.

Congruent

Equal Measures

Scale Factor

The ratio of the length/size of the drawing to the length/size of the actual object

Scale

The ratio of two measurements

Perimeter

The measurement around a shape

Circumference

The perimeter around a circle

Radius

A line segment that goes from the center of the circle to any point on the perimeter

Diameter

A line segment that goes through the center of a circle

Acute Triangle

A triangle with three acute angles

Obtuse Triangle

A triangle with one obtuse angle and two acute angles

Right Triangle

Triangle with one 90 degree angle

The sum of angle measures in a triangle

180 degrees

The sum of the angle measure in a quarilateral

360 degrees

Area of a circle

A=pi times r squared

Perimeter of a semi-circle

C=(2 x pi x r)/2

Area of a Triangle

A = 1/2 x b x h

Circumference of a circle

2 x pii x r

A point

An exact position on a plane surface

A line

Set of points in a straight path, extending forever in both directions

Line segment

A part of a line between two definite endpoints.

Ray

A part of a line that has one endpoint and extends forever in one direction

Vertex

A point where two or more rays or line segments meet

Angle

Figure formed by two rays with a common endpoint (vertex)

The sum of angle measures around a point

360 degrees

Parallel lines

Lines that never meet

Perpendicular lines

Lines that intersect at right angles

cross-section

the face that results when cutting through a solid shape

pi

3.14

regular polygon

a polygon with all sides and all angles equal

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