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Name———————————————————————— Lesson
4.1
Date —————————————
Study Guide
For use with the lesson “Apply Triangle Sum Properties”
goal
Classify triangles and find measures of their angles.
Vocabulary
A triangle is a polygon with three sides.
A scalene triangle has no congruent sides.
An isosceles triangle has at least two congruent sides.
An equilateral triangle has three congruent sides.
An acute triangle has three acute angles.
A right triangle has one right angle.
An obtuse triangle has one obtuse angle.
When the sides of a polygon are extended, other angles are formed.
The original angles are the interior angles. The angles that form linear
pairs with the interior angles are the exterior angles.
Lesson 4.1
An equiangular triangle has three congruent angles.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Theorem 4.1 Triangle Sum Theorem: The sum of the measures of the
interior angles of a triangle is 1808.
Theorem 4.2 Exterior Angle Theorem: The measure of an exterior
angle of a triangle is equal to the sum of the measures of the two
nonadjacent interior angles.
Corollary to the Triangle Sum Theorem: The acute angles of a right
triangle are complementary.
example 1
Classify triangles by sides and by angles
Classify the triangle by its sides and by its angles.
a.
E
19
D
198
1108
23
B
b.
8
518
458
F
4 2
4
A
4
458
C
Solution
a. Triangle DEF has one obtuse angle and no congruent sides.
So, n DEF is an obtuse scalene triangle.
b. Triangle ABC has one right angle and two congruent sides.
So, n ABC is a right isosceles triangle.
Geometry
Chapter Resource Book
4-13
Name———————————————————————— Lesson
4.1
Date —————————————
Study Guide continued
For use with the lesson “Apply Triangle Sum Properties”
Exercises for Example 1
Classify the triangle by its sides and by its angles.
1.
2.
3.
608
5
308
17
5.8
308
18
608
17
Find angle measures
a. Find m∠ BAC and m∠ BCA.
B
b. Find m∠ BCD and m∠ ABC.
C
B
(4x 2 5)8
2x8
768
(3x 1 11)8
A
A
(5x 2 2)8
C
D
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Lesson 4.1
example 2
10
308
17
608
608
2.9
1208
10
Solution
a. (4x 2 5)8 1 (3x 1 11)8 5 908Use Corollary to the Triangle
Sum Theorem.
x 5 12
Solve for x.
So, m∠ BCA 5 (4x 2 5)8 5 (4 p 12 2 5)8 5 438 and
m∠ BAC 5 (3x 1 11)8 5 (3 p 12 1 11)8 5 478.
Use Exterior Angle Theorem.
b. (5x 2 2)8 5 2x8 1 768
x 5 26
Solve for x.
So, m∠ BCD 5 (5x 2 2)8 5 (5 p 26 2 2)8 5 1288 and
m∠ ABC 5 2x8 5 2(26)8 5 528.
Exercises for Example 2
4. Find m∠ ABD and m∠ BDC.
A
B
3x 8
2x 8
D
4-14
Geometry
Chapter Resource Book
308
5. Find m∠ CAB and m∠ CBA.
C
A
C
8x8
x8
B
Answers for Chapter 4
Lesson 4.1 Apply Triangle Sum
Properties
Teaching Guide
19. 1228 20. 388 21. m∠ A 5 608, m∠ B 5 308,
m∠ C 5 908 22. m∠ A 5 608, m∠ B 5 308,
m∠ C 5 908
23. 60, 30 24. 45, 51 25. 24, 66
26. scalene; right
Practice Level C
1. scalene; 20; acute 2. isosceles; 25; acute
3. equilateral; 60; equiangular 4. x 5 10; y 5 71
Technology Activity
5. x 5 50; y 5 33 6. x 5 15; y 5 42
1. The angle measures change; the sum of the
angle measures stays the same.
2. Sample answer: 5 triangles; the sum of the
angle measures is 1808. 3. The sum of the
measures of the angles of any triangle is 1808.
Practice Level A
1. equilateral 2. isosceles 3. scalene
7. x 5 85; y 5 58 8. 608 9. 1208 10. 608
11. 1208 12. 308 13. 308 14. 748
15. m∠ B 5 1158; m∠ C 5 238 16. 8 in. by
11 in. by 11 in.; 8 in. by 8 in. by 14 in.
17. 864 in. 18. 576 in. 19. 1344 in.
Study Guide
4. isosceles 5. right 6. acute 7. obtuse
1. right scalene 2. equiangular equilateral
8. equiangular 9. 55 10. 35 11. 45 12. 158
13. 90 14. 62 15. m∠ 1 5 608
16. m∠ 1 5 608; m∠ 2 5 308; m∠ 3 5 808
18. m∠ A 5 458; m∠ B 5 908; m∠ C 5 458
19. scalene; obtuse
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
3. obtuse isosceles 4. m∠ ABD 5 908,
m∠ BDC 5 608 5. m∠ CAB 5 808,
m∠ CBA 5 108
Problem Solving Workshop:
Worked Out Example
}
}
}
1. The legs are ​RS​ and TS​
​  , and the base is RT​
​ ; 
17. m∠ 1 5 708; m∠ 2 5 658; m∠ 3 5 958
m∠ 1 5 55°, m∠ 2 5 110°
}
}
2. The legs are ​RS​ and TS​
​  , and the hypotenuse
}
is RT​
​ ;  m∠ SRT 5 54°, m∠ 1 5 144°
Practice Level B
1. sometimes 2. never 3. never 4. sometimes
5. scalene, obtuse 6. scalene, right
Challenge Practice
7. isosceles, acute
8.
y
9.
1. (5, 9)
y
y
2. (9, 1)
C (5, 9)
y
C
B
2
A
B(7, 5)
2 A
C
x
2
B(7, 5)
2
answers
1. Check drawings. 2. Check drawings. It is
impossible to construct triangles with 3 congruent
sides and 1 right angle and with 3 congruent sides
and 1 obtuse angle.
Congruent Triangles
B
A(3, 3)
A(3, 3)
x
D(9, 1)
x
scalene; right trianglescalene; not a right
triangle
y
10.
isosceles; not a right triangle
x
3. 608, 608, 608 4. 638, 368, 818; acute
5. x 5 29, y 5 64 6. x 5 12.9, y 5 51.3
7. m∠ A 5 m∠ 1
C
8. GIVEN: n ABC
2
B
A
2
x
11. 30; right 12. 25; acute 13. 120; acute
14. 1318 15. 1008 16. 1258 17. 368 18. 1228
PROVE: S
um of exterior angles of n ABC
is 3608.
By the definition of a straight angle, you know that
m∠ 1 1 m∠ 2 5 1808, m∠ 3 1 m∠ 4 5 1808, and
m∠ 5 1 m∠ 6 5 1808. So, it follows that the sum
of all six angles is 1808 1 1808 1 1808 5 5408.
Geometry
Chapter Resource Book
CS10_CC_G_MECR710761_C4AK.indd 45
A45
4/28/11 6:14:06 PM