1. 4-1 Triangles and Angles Warm Up Lesson Presentation Lesson Quiz
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz

2. 4.1 Triangles and Angles Warm Up
Classify each angle as acute, obtuse, or right. 3.
4. If the perimeter is 47, find x and the lengths of the three sides.
right
acute
obtuse
x = 5; 8; 16; 23

3. Objectives 4.1 Triangles and Angles
Classify triangles by their angle measures and side lengths.
Use triangle classification to find angle measures and side lengths. Find the measures of interior and exterior angles of triangles.
Apply theorems about the interior and exterior angles of triangles.

4. Vocabulary 4.1 Triangles and Angles acute triangle Corollary
equiangular triangle Legs
right triangle Adjacent
obtuse triangle Exterior
equilateral triangle Interior
isosceles triangle Hypotenuse
scalene triangle base

5. 4.1 Triangles and Angles
Recall that a triangle ( ) is a polygon with three sides. Triangles can be classified in two ways: by their angle measures or by their side lengths.

6. 4.1 Triangles and Angles C A B AB, BC, and AC are the sides of ABC.
A, B, C are the triangle's vertices.

7. 4.1 Triangles and Angles In a triangle, two sides sharing a common
vertex are Adjacent sides.

8. 4.1 Triangles and Angles
In a right triangle, the two sides making the right angle are called Legs.
The side opposite of the right angle is called the Hypothenuse.

9. 4.1 Triangles and Angles
In an isosceles triangle, the two sides that are congruent are called the Legs.
The third side is the called the base.

10. 4.1 Triangles and Angles Acute Triangle Three acute angles
Triangle Classification
By Angle Measures
Acute Triangle
Three acute angles

11. Three congruent acute angles
4.1 Triangles and Angles
Triangle Classification
By Angle Measures
Equiangular Triangle
Three congruent acute angles

12. 4.1 Triangles and Angles Right Triangle One right angle
Triangle Classification
By Angle Measures
Right Triangle
One right angle

13. 4.1 Triangles and Angles Obtuse Triangle One obtuse angle
Triangle Classification
By Angle Measures
Obtuse Triangle
One obtuse angle

14. Example 1A: Classifying Triangles by Angle Measures
4.1 Triangles and Angles
Example 1A: Classifying Triangles by Angle Measures
Classify BDC by its angle measures.
B is an obtuse angle.
B is an obtuse angle. So BDC is an obtuse triangle.

15. Example 1B: Classifying Triangles by Angle Measures
4.1 Triangles and Angles
Example 1B: Classifying Triangles by Angle Measures
Classify ABD by its angle measures.
ABD and CBD form a linear pair, so they are supplementary.
Therefore mABD + mCBD = 180°. By substitution, mABD + 100° = 180°. So mABD = 80°. ABD is an acute triangle by definition.

16. 4.1 Triangles and Angles Check It Out! Example 1
Classify FHG by its angle measures.
EHG is a right angle. Therefore mEHF +mFHG = 90°. By substitution, 30°+ mFHG = 90°. So mFHG = 60°.
FHG is an equiangular triangle by definition.

17. 4.1 Triangles and Angles Equilateral Triangle Three congruent sides
Triangle Classification
By Side Lengths
Equilateral Triangle
Three congruent sides

18. At least two congruent sides
4.1 Triangles and Angles
Triangle Classification
By Side Lengths
Isosceles Triangle
At least two congruent sides

19. 4.1 Triangles and Angles Scalene Triangle No congruent sides
Triangle Classification
By Side Lengths
Scalene Triangle
No congruent sides

20. 4.1 Triangles and Angles Remember!
When you look at a figure, you cannot assume segments are congruent based on appearance. They must be marked as congruent.

21. 4.1 Triangles and Angles
When the sides of the triangle are extended, the original angles are the interior angles.
The angles that are adjacent (next to) the interior angles are the exterior angles.

22. 4.1 Triangles and Angles

23. 4.1 Triangles and Angles
An auxiliary line is a line that is added to a figure to aid in a proof.
An auxiliary line used in the Triangle Sum Theorem

24. 4.1 Triangles and Angles

25. 4.1 Triangles and Angles
A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.

26. 4.1 Triangles and Angles
An auxiliary line is a line that is added to a figure to aid in a proof.
An auxiliary line used in the Triangle Sum Theorem

27. Example 1A: Application
4.1 Triangles and Angles
Example 1A: Application
After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mXYZ.
mXYZ + mYZX + mZXY = 180°
Sum. Thm
Substitute 40 for mYZX and 62 for mZXY.
mXYZ = 180
mXYZ = 180
Simplify.
mXYZ = 78°
Subtract 102 from both sides.

28. Example 1B: Application
4.1 Triangles and Angles
Example 1B: Application
After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mYWZ.
118°
Step 1 Find mWXY.
mYXZ + mWXY = 180°
Lin. Pair Thm. and  Add. Post.
62 + mWXY = 180
Substitute 62 for mYXZ.
mWXY = 118°
Subtract 62 from both sides.

29. Example 1B: Application Continued
4.2 Congruence and Triangles
Example 1B: Application Continued
After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mYWZ.
118°
Step 2 Find mYWZ.
mYWX + mWXY + mXYW = 180°
Sum. Thm
Substitute 118 for mWXY and 12 for mXYW.
mYWX = 180
mYWX = 180
Simplify.
mYWX = 50°
Subtract 130 from both sides.

30. 4.1 Triangles and Angles Check It Out! Example 1
Use the diagram to find mMJK.
mMJK + mJKM + mKMJ = 180°
Sum. Thm
Substitute 104 for mJKM and 44 for mKMJ.
mMJK = 180
mMJK = 180
Simplify.
mMJK = 32°
Subtract 148 from both sides.

31. 4.1 Triangles and Angles
A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.

32. Example 2: Finding Angle Measures in Right Triangles
4.1 Triangles and Angles
Example 2: Finding Angle Measures in Right Triangles
One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle?
Let the acute angles be A and B, with mA = 2x°.
mA + mB = 90°
Acute s of rt. are comp.
2x + mB = 90
Substitute 2x for mA.
mB = (90 – 2x)°
Subtract 2x from both sides.

33. 4.1 Triangles and Angles Check It Out! Example 2a
The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle?
Let the acute angles be A and B, with mA = 63.7°.
mA + mB = 90°
Acute s of rt. are comp.
mB = 90
Substitute 63.7 for mA.
mB = 26.3°
Subtract 63.7 from both sides.

34. 4.1 Triangles and Angles Check It Out! Example 2b
The measure of one of the acute angles in a right triangle is x°. What is the measure of the other acute angle?
Let the acute angles be A and B, with mA = x°.
mA + mB = 90°
Acute s of rt. are comp.
x + mB = 90
Substitute x for mA.
mB = (90 – x)°
Subtract x from both sides.

35. 4.2 Congruence and Triangles
Check It Out! Example 2c
The measure of one of the acute angles in a right triangle is What is the measure of the other acute angle?
2° 5
Let the acute angles be A and B, with mA =
2° 5
mA + mB = 90°
Acute s of rt. are comp.
mB = 90
2 5
Substitute for mA.
2 5
mB = 41
3° 5
Subtract from both sides.
2 5

36. 4.2 Congruence and Triangles
The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure.
Exterior
Interior

37. 4.1 Triangles and Angles
An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and extension of an adjacent side.
4 is an exterior angle.
Exterior
Interior
3 is an interior angle.

38. 4.1 Triangles and Angles
Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle.
4 is an exterior angle.
The remote interior angles of 4 are 1 and 2.
Exterior
Interior
3 is an interior angle.

39. 4.1 Triangles and Angles

40. Example 3: Applying the Exterior Angle Theorem
4.1 Triangles and Angles
Example 3: Applying the Exterior Angle Theorem
Find mB.
mA + mB = mBCD
Ext.  Thm.
Substitute 15 for mA, 2x + 3 for mB, and 5x – 60 for mBCD.
15 + 2x + 3 = 5x – 60
2x + 18 = 5x – 60
Simplify.
Subtract 2x and add 60 to both sides.
78 = 3x
26 = x
Divide by 3.
mB = 2x + 3 = 2(26) + 3 = 55°

41. 4.2 Congruence and Triangles
Check It Out! Example 3
Find mACD.
mACD = mA + mB
Ext.  Thm.
Substitute 6z – 9 for mACD, 2z + 1 for mA, and 90 for mB.
6z – 9 = 2z
6z – 9 = 2z + 91
Simplify.
Subtract 2z and add 9 to both sides.
4z = 100
z = 25
Divide by 4.
mACD = 6z – 9 = 6(25) – 9 = 141°

42. Example 2A: Classifying Triangles by Side Lengths
4.1 Triangles and Angles
Example 2A: Classifying Triangles by Side Lengths
Classify EHF by its side lengths.
From the figure, So HF = 10, and EHF is isosceles.

43. Example 2B: Classifying Triangles by Side Lengths
4.1 Triangles and Angles
Example 2B: Classifying Triangles by Side Lengths
Classify EHG by its side lengths.
By the Segment Addition Postulate, EG = EF + FG = = 14. Since no sides are congruent, EHG is scalene.

44. 4.1 Triangles and Angles Check It Out! Example 2
Classify ACD by its side lengths.
From the figure, So AC = 15, and ACD is isosceles.

45. Example 3: Using Triangle Classification
4.1 Triangles and Angles
Example 3: Using Triangle Classification
Find the side lengths of JKL.
Step 1 Find the value of x.
Given.
JK = KL
Def. of  segs.
Substitute (4x – 10.7) for JK and (2x + 6.3) for KL.
4x – 10.7 = 2x + 6.3
Add 10.7 and subtract 2x from both sides.
2x = 17.0
x = 8.5
Divide both sides by 2.

46. 4.1 Triangles and Angles Example 3 Continued
Find the side lengths of JKL.
Step 2 Substitute 8.5 into the expressions to find the side lengths.
JK = 4x – 10.7
= 4(8.5) – 10.7 = 23.3
KL = 2x + 6.3
= 2(8.5) = 23.3
JL = 5x + 2
= 5(8.5) + 2 = 44.5

47. 4.1 Triangles and Angles Check It Out! Example 3
Find the side lengths of equilateral FGH.
Step 1 Find the value of y.
Given.
FG = GH = FH
Def. of  segs.
Substitute
(3y – 4) for FG and (2y + 3) for GH.
3y – 4 = 2y + 3
Add 4 and subtract 2y from both sides.
y = 7

48. Check It Out! Example 3 Continued
4.1 Triangles and Angles
Check It Out! Example 3 Continued
Find the side lengths of equilateral FGH.
Step 2 Substitute 7 into the expressions to find the side lengths.
FG = 3y – 4
= 3(7) – 4 = 17
GH = 2y + 3
= 2(7) + 3 = 17
FH = 5y – 18
= 5(7) – 18 = 17

49. 4.1 Triangles and Angles Example 4: Application
A steel mill produces roof supports by welding pieces of steel beams into equilateral triangles. Each side of the triangle is 18 feet long. How many triangles can be formed from 420 feet of steel beam?
The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle.
P = 3(18)
P = 54 ft

50. Example 4: Application Continued
4.1 Triangles and Angles
Example 4: Application Continued
A steel mill produces roof supports by welding pieces of steel beams into equilateral triangles. Each side of the triangle is 18 feet long. How many triangles can be formed from 420 feet of steel beam?
To find the number of triangles that can be made from 420 feet of steel beam, divide 420 by the amount of steel needed for one triangle.
420  54 = 7 triangles
7 9
There is not enough steel to complete an eighth triangle. So the steel mill can make 7 triangles from a 420 ft. piece of steel beam.

51. 4.1 Triangles and Angles Check It Out! Example 4a
Each measure is the side length of an equilateral triangle. Determine how many 7 in. triangles can be formed from a 100 in. piece of steel.
The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle.
P = 3(7)
P = 21 in.

52. Check It Out! Example 4a Continued
4.1 Triangles and Angles
Check It Out! Example 4a Continued
Each measure is the side length of an equilateral triangle. Determine how many 7 in. triangles can be formed from a 100 in. piece of steel.
To find the number of triangles that can be made from 100 inches of steel, divide 100 by the amount of steel needed for one triangle.
100  7 = 14 triangles
2 7
There is not enough steel to complete a fifteenth triangle. So the manufacturer can make 14 triangles from a 100 in. piece of steel.

53. 4.1 Triangles and Angles Check It Out! Example 4b
Each measure is the side length of an equilateral triangle. Determine how many 10 in. triangles can be formed from a 100 in. piece of steel.
The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle.
P = 3(10)
P = 30 in.

54. Check It Out! Example 4b Continued
4.1 Triangles and Angles
Check It Out! Example 4b Continued
Each measure is the side length of an equilateral triangle. Determine how many 10 in. triangles can be formed from a 100 in. piece of steel.
To find the number of triangles that can be made from 100 inches of steel, divide 100 by the amount of steel needed for one triangle.
100  10 = 10 triangles
The manufacturer can make 10 triangles from a 100 in. piece of steel.

55. 4.1 Triangles and Angles Lesson Quiz I 1. MNQ 2. NQP
Classify each triangle by its angles and sides.
MNQ
NQP
MNP
4. Find the side lengths of the triangle.
acute; equilateral
obtuse; scalene
acute; scalene
29; 29; 23

56. 4.1 Triangles and Angles Lesson Quiz: Part II
5. The measure of one of the acute angles in a right triangle is 56 °. What is the measure of the other acute angle?
6. Find mABD.
2 3
33 °
1 3
124°

57. 4.1 Triangles and Angles Lesson Quiz: Part III
7. The diagram is a map showing John's house, Kay's house, and the grocery store. What is the angle the two houses make with the store?
30°