Download State whether each sentence is true or false . If false

Study Guide and Review
State whether each sentence is true or false . If false , replace the underlined word or phrase to make a
true sentence.
1. An equiangular triangle is also an example of an acute triangle.
SOLUTION: true; all 3 angles of an equiangular triangle are acute and is this an example of an acute triangle
2. A triangle with an angle that measures greater than 90° is a right triangle.
SOLUTION: false; an angle that measures greater than 90 is not right it is obtuse
3. An equilateral triangle is always equiangular.
SOLUTION: true; if all sides of the triangle are congruent then all angles are also congruent
4. A scalene triangle has at least two congruent sides.
SOLUTION: false; a triangle that has two congruent sides is an isosceles triangle
5. The vertex angles of an isosceles triangle are congruent.
SOLUTION: false; an isosceles triangle's base angles congruent
6. An included side is the side located between two consecutive angles of a polygon.
SOLUTION: true; an included side is the side located between two consecutive angles of a polygon
7. The three types of congruence transformations are rotation, reflection, and translation.
SOLUTION: true; rotations, reflections, and translations are the three types of congruence transformations
8. A rotation moves all points of a figure the same distance and in the same direction.
SOLUTION: false; the congruence transformation that moves all points of a figure the same distance in the same direction is a
translation
9. A flow proof uses figures in the coordinate plane and algebra to prove geometric concepts.
SOLUTION: false; the proof that uses figures in the coordinate plane and algebra to prove geometric concepts is called a
coordinate proof
10. The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
SOLUTION: true; an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles
Classify each triangle as acute, equiangular, obtuse, or right.
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coordinate proof
10. The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.
SOLUTION: Study
Guide and Review
true; an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles
Classify each triangle as acute, equiangular, obtuse, or right.
11. SOLUTION: is obtuse, since
.
12. SOLUTION: is a right triangle, since
13. SOLUTION: is a right triangle, since
ALGEBRA Find x and the measures of the unknown sides of each triangle.
14. SOLUTION: Here, RS = RT.
So,
Solve for x.
Substitute
.
in RS.
Since RS = RT, RT = 31.
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Study
Guide
Since
RS and
= RT,Review
RT = 31.
15. SOLUTION: Since the triangle is an equilateral triangle, set two of the sides equal to each other.
Substitute x = 6 in for x for each of the sides.
JL: KL: JK: 16. MAPS The distance from Chicago to Cleveland to Cincinnati and back to Chicago is 900 miles. The distance from
Chicago to Cleveland is 50 miles more than the distance from Cincinnati to Chicago, and the distance from Cleveland
to Cincinnati is 50 miles less than the distance from Cincinnati to Chicago. Find each distance and classify the
triangle formed by the three cities.
SOLUTION: Let x be the distance from Cincinnati to Chicago. So, distance from Chicago to Cleveland is x + 50 and the distance
from Cleveland to Cincinnati is x – 50. Form a triangle for the given information.
Here,
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Substitute x = 300 in x + 50.
x + 50 = 350
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JK: Study Guide and Review
16. MAPS The distance from Chicago to Cleveland to Cincinnati and back to Chicago is 900 miles. The distance from
Chicago to Cleveland is 50 miles more than the distance from Cincinnati to Chicago, and the distance from Cleveland
to Cincinnati is 50 miles less than the distance from Cincinnati to Chicago. Find each distance and classify the
triangle formed by the three cities.
SOLUTION: Let x be the distance from Cincinnati to Chicago. So, distance from Chicago to Cleveland is x + 50 and the distance
from Cleveland to Cincinnati is x – 50. Form a triangle for the given information.
Here,
Substitute x = 300 in x + 50.
x + 50 = 350
Substitute x = 300 in x – 50.
x – 50 = 250
So, the distance from Cincinnati to Chicago is 300 miles, the distance from Chicago to Cleveland is 350 miles, and the
distance from Cleveland to Cincinnati is 250 miles.
No two sides have equal lengths, so it is scalene.
Find the measure of each numbered angle.
17. 1
SOLUTION: The sum of the measures of the angles of a triangle is 180.
Here,
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Study Guide and Review
No two sides have equal lengths, so it is scalene.
Find the measure of each numbered angle.
17. 1
SOLUTION: The sum of the measures of the angles of a triangle is 180.
Here,
18. 2
SOLUTION: By the Exterior Angle Theorem, So,
.
19. 3
SOLUTION: By the Exterior Angle Theorem, So,
.
Again use the Exterior Angle Theorem to find
Substitute
.
.
That is,
20. HOUSES The roof support on Lamar’s house is in the shape of an isosceles triangle with base angles of 38°. Find x.
SOLUTION: The sum of the measures of the angles of a triangle is 180.
Here,
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Show that the polygons are congruent by identifying all congruent corresponding parts. Then write a
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Study Guide and Review
That is,
20. HOUSES The roof support on Lamar’s house is in the shape of an isosceles triangle with base angles of 38°. Find x.
SOLUTION: The sum of the measures of the angles of a triangle is 180.
Here,
Show that the polygons are congruent by identifying all congruent corresponding parts. Then write a
congruence statement.
21. SOLUTION: Angles:
Sides:
All corresponding parts of the two polygons are congruent. Therefore, polygon ABCD
polygon FGHJ. 22. SOLUTION: Angles:
Sides:
.
All corresponding parts of the two triangles are congruent. Therefore, ΔXYZ
ΔJKL.
23. MOSAIC TILING A section of a mosaic tiling is shown. Name the triangles that appear to be congruent.
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SOLUTION: The figure looks to be a square in the middle with a triangle attached to each of the four sides. Each one of those
SOLUTION: Angles:
Sides:
.
Study
Guide and Review
All corresponding parts of the two triangles are congruent. Therefore, ΔXYZ
ΔJKL.
23. MOSAIC TILING A section of a mosaic tiling is shown. Name the triangles that appear to be congruent.
SOLUTION: The figure looks to be a square in the middle with a triangle attached to each of the four sides. Each one of those
four outer triangles appear to be congruent. Now look at the square in the center. The square is also made up of four
triangles. Each one of those four interior triangles appear to becongruent.
Determine whether
Explain.
24. A(5, 2), B(1, 5), C(0, 0), X(–3, 3), Y(–7, 6), Z(–8, 1)
SOLUTION: Use the Distance Formula to find the lengths of
.
has endpoints A(5, 2) and B(1, 5).
Substitute.
has endpoints B(1, 5) and C(0, 0).
Substitute.
has endpoints C(0, 0) and A(5, 2).
Substitute.
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has endpoints C(0, 0) and A(5, 2).
Study Guide and Review
Substitute.
Similarly, find the lengths of
.
has endpoints X(–3, 3) and Y(–7, 6).
Substitute.
has endpoints Y(–7, 6) and Z(–8, 1).
Substitute.
has endpoints Z(–8, 1) and X(–3, 3).
Substitute.
So,
.
Each pair of corresponding sides has the same measure so they are congruent.
by SSS.
25. A(3, –1), B(3, 7), C(7, 7), X(–7, 0), Y(–7, 4), Z(1, 4)
SOLUTION: Use the Distance Formula to find the lengths of
.
has endpoints A(3, –1) and B(3, 7).
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Substitute.
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Each pair of corresponding sides has the same measure so they are congruent.
25. A(3, –1), B(3, 7), C(7, 7), X(–7, 0), Y(–7, 4), Z(1, 4)
Study Guide and Review
SOLUTION: Use the Distance Formula to find the lengths of
by SSS.
.
has endpoints A(3, –1) and B(3, 7).
Substitute.
has endpoints B(3, 7) and C(7, 7).
Substitute.
has endpoints C(7, 7) and A(3,–1).
Substitute.
Similarly, find the lengths of
.
has endpoints X(–7, 0) and Y(–7, 4).
Substitute.
has endpoints Y(–7, 4) and Z(1, 4).
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Substitute.
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Study Guide and Review
has endpoints Y(–7, 4) and Z(1, 4).
Substitute.
has endpoints Z(1, 4) and X(–7, 0).
Substitute.
The triangles are not congruent, since the corresponding sides of the two triangles are not congruent.
Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to
prove that they are congruent, write not possible.
26. SOLUTION: The two triangles share a side in common and by the reflexive property that side is congruent to itself. The other leg
of the two triangles are marked congruent. The included angle in one of the triangles is marked as a right angle. The
other included angle is also right because they form a linear pair. The two triangles are congruent by the SAS
postulate.
27. SOLUTION: The two triangles share a side and by the reflexive property that side is congruent to itself. However; we cannot
prove any other sides or angles congruent with the information given. It is not possible to prove the triangles
congruent.
28. PARKS The diagram shows a park in the shape of a pentagon with five sidewalks of equal length leading to a
central point. If all the angles at the central point have the same measure, how could you prove that
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SOLUTION: The two triangles share a side and by the reflexive property that side is congruent to itself. However; we cannot
prove
anyand
otherReview
sides or angles congruent with the information given. It is not possible to prove the triangles
Study
Guide
congruent.
28. PARKS The diagram shows a park in the shape of a pentagon with five sidewalks of equal length leading to a
central point. If all the angles at the central point have the same measure, how could you prove that
SOLUTION: All sidewalks are of equal length so segment AX = segment BX = segment CX = segment DX. The included angles
of both triangles are also congruent as given. The triangles have two sides and the included angle congruent. The
triangles are congruent by SAS.
Write a two-column proof.
29. Given:
Prove:
SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here,
you are given that two line segments, of a figure, are parallel and congruent. Use what you know about parallel lines
and alternate interior angles to walk through the proof and prove the two triangles congruent. Statements (Reasons)
1.
(Given)
2.
(Alternate Interior s Thm.)
3.
(Given)
4.
(Alternate Interior s Thm.)
5.
(ASA)
30. KITES Denise’s kite is shown in the figure below. Given that
bisects both XWZ and XYZ, prove that
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SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here
2.
(Alternate Interior
3.
(Given)
4.
(Alternate Interior
Study
5. Guide and Review
(ASA)
s Thm.)
s Thm.)
30. KITES Denise’s kite is shown in the figure below. Given that
bisects both XWZ and XYZ, prove that
SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here
you are given one segment is an angle bisector of two different angles. Use what you know about angle bisectors to
walk through the proof and prove that the two triangles are congruent.
Statements (Reasons)
1.
bisects both XWY and XZY. (Given)
2. XWY ZWY (Def. of Bisector)
3.
(Reflexive Property)
4. XYW ZYW (Def. of Bisector)
5.
(ASA)
Find the value of each variable.
31. SOLUTION: The given triangle has three congruent angles, so it is equilateral. So, all the sides are equal in length.
32. SOLUTION: The given triangle has two congruent sides, so it is isosceles. Therefore, the base angles are congruent. The
measures of the base angles are 52 each. We know that the sum of the measures of the angles of a triangle is 180.
Here,
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32. SOLUTION: The given triangle has two congruent sides, so it is isosceles. Therefore, the base angles are congruent. The
measures of the base angles are 52 each. We know that the sum of the measures of the angles of a triangle is 180.
Here,
33. PAINTING Pam is painting using a wooden easel. The support bar on the easel forms an isosceles triangle with the
two front supports. According to the figure below, what are the measures of the base angles of the triangle?
SOLUTION: Since the triangle is isosceles, the base angles are congruent. Let x be the measure of each base angle. We know
that the sum of the measures of the angles of a triangle is 180. So,
Solve for x.
Therefore, the measures of the base angles are 77.5 each.
Identify the type of congruence transformation shown as a reflection, translation, or rotation.
34. SOLUTION: This transformation has moved all points on the original figure the same distance in the same direction. This is a
translation.
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34. SOLUTION: This transformation has moved all points on the original figure the same distance in the same direction. This is a
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Guide and Review
translation.
35. SOLUTION: The transformation has been flipped over the x-axis. Each point on the preimage, and each point on the image, are
the same distance from the x-axis. This is a reflection.
36. SOLUTION: The transformation has been flipped over the y-axis. Each point on the preimage, and each point on the image, are
the same distance from the y-axis. This is a reflection.
37. SOLUTION: The transformation has been turned around a fixed point, in this case the origin, through a specific angle, and in a
specific direction. Each point of the original figure and its image are the same distance from the center. This is a
rotation.
38. Triangle ABC with vertices A(1, 1), B(2, 3), and C(3, –1) is a transformation of
with vertices M (–1, 1), N(–
2, 3), and O(–3, –1). Graph the original figure and its image. Identify the transformation and verify that it is a
congruence transformation.
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Each vertex and its image are the same distance from the y-axis. This is a reflection. That is,
of
.
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is a reflection 38. Triangle ABC with vertices A(1, 1), B(2, 3), and C(3, –1) is a transformation of
with vertices M (–1, 1), N(–
2, 3), and O(–3, –1). Graph the original figure and its image. Identify the transformation and verify that it is a
Study
Guide and
Review
congruence
transformation.
SOLUTION: Each vertex and its image are the same distance from the y-axis. This is a reflection. That is,
of
.
has endpoints A(1, 1) and B(2, 3).
Use the distance formula.
is a reflection has endpoints B(2, 3) and C(3, –1).
has endpoints C(3, –1) and A(1, 1).
Similarly, find the lengths of
.
has endpoints M (–1, 1) and N(–2, 3).
has endpoints
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3) and
O(–3, –1).
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Study Guide and Review
has endpoints N(–2, 3) and O(–3, –1).
has endpoints O(–3, –1) and M (–1, 1).
So,
.
Each pair of corresponding sides has the same measure so they are congruent.
congruence transformation.
by SSS. So, it is a Position and label each triangle on the coordinate plane.
39. right
with right angle at point M and legs of lengths a and 2a.
SOLUTION: Since this is a right triangle, two sides can be located on axis. Placing the right angle of the triangle,
, at the
origin will allow the two legs to be along the x- and y-axes. Position the triangle in the first quadrant. Since N is on
the y-axis, its x-coordinate is 0. Its y-coordinate is a because the leg is a units long. Since O is on the x-axis, its ycoordinate is 0. Its x-coordinate is 2a because the leg is 2a units long.
The coordinates of O are (2a, 0).
40. isosceles
with height h and base
with length 2a.
SOLUTION: has two congruent sides, so it is isosceles. Point W is at origin, so its coordinates are (0, 0). Point Y is on the
x-axis, so its y-coordinate is 0. Here WY = 2a. So, the x-coordinate of Y is 2a. The coordinates of Y are (2a, 0). Find
the coordinates of point X , that makes the triangle isosceles. This point is half way between W and Y, so the xcoordinate is a. Since the height is h, its y-coordinate is h. So, the coordinates of X are (a, h).
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Study Guide and Review
40. isosceles
with height h and base
with length 2a.
SOLUTION: has two congruent sides, so it is isosceles. Point W is at origin, so its coordinates are (0, 0). Point Y is on the
x-axis, so its y-coordinate is 0. Here WY = 2a. So, the x-coordinate of Y is 2a. The coordinates of Y are (2a, 0). Find
the coordinates of point X , that makes the triangle isosceles. This point is half way between W and Y, so the xcoordinate is a. Since the height is h, its y-coordinate is h. So, the coordinates of X are (a, h).
41. GEOGRAPHY Jorge plotted the cities of Dallas, San Antonio , and Houston as shown. Write a coordinate proof to
show that the triangle formed by these cities is scalene.
SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here,
you are given the coordinates for the three vertices of a triangle. If no two sides of the triangle are congruent, then
the triangle is scalene. Use the Distance Formula and a calculator to find the distance between each location. Walk
through the proof and prove the triangle is scalene.
Statements (Reasons)
1. D(8, 28), S(0, 0), and H(19, 7) (Given)
2. DS =
(Distance Formula)
3. SH =
(Distance Formula)
(Distance Formula)
4. DH =
5.
is scalene. (Definition of scalene)
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