1. 4-2: Measuring Angles in Triangles
Expectations:
G1.2.1: Prove that the sum of the angle measures of a triangle is 180 and that the measure of an exterior angle of a triangle is the sum of the measures of the 2 remote interior angles.
G1.2.2: Solve problems involving angle measure and side length of triangles.
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4-2: Measuring Angles in Triangles

2. 4-2: Measuring Angles in Triangles
Linear Triples
∠A, ∠B and ∠C form a linear triple iff ∠A is adjacent to ∠B, ∠B is adjacent to ∠C, ∠A and ∠C have no points in common (other than the vertex) and the sum of their measures is 180. ∠A ∠B ∠C
∠A, ∠B and ∠C form a linear triple.
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4-2: Measuring Angles in Triangles

3. Justify the sum of the measures of ∠A, ∠B and ∠C.
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4-2: Measuring Angles in Triangles

4. 4-2: Measuring Angles in Triangles
Triangle Sum Theorem:
The sum of the measures of the angles of a triangle is _______°.
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4-2: Measuring Angles in Triangles

5. Determine the value of x.C x A
x - 8 B 58
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4-2: Measuring Angles in Triangles

6. 4-2: Measuring Angles in Triangles
In triangle ABC below, the measure of ∠A is 20° and the measure of ∠B is 3 times larger than the measure of ∠C. What is the measure of angle B?
40°
60°
80°
120°
160° B C A
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4-2: Measuring Angles in Triangles

7. Calculate the measures of each angle below.
8x-12
7x+3
5x+9 A B
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4-2: Measuring Angles in Triangles

8. 4-2: Measuring Angles in Triangles
If ∠A≅∠D and ∠C≅∠F, what may you conclude about ∠B and ∠E?
Can you justify your conjecture? C F A B D E
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9. 4-2: Measuring Angles in Triangles
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10. 4-2: Measuring Angles in Triangles
Third Angle Congruence Theorem:
If two angles of one triangle are _____________ to 2 angles of a second triangle, then the third angles are also _____________ .
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4-2: Measuring Angles in Triangles

11. 4-2: Measuring Angles in Triangles
Exterior Angles
An angle is an exterior angle of a triangle iff it is the union of one side of the triangle and the extension of the other side that shares the same vertex as the first side.
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12. 4-2: Measuring Angles in Triangles
Exterior Angles C
∠CBD is an exterior angle for ∆ABC A B D
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13. 4-2: Measuring Angles in Triangles
Exterior Angles C
∠ABE is an exterior angle for ∆ABC A B E
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14. 4-2: Measuring Angles in Triangles
Exterior Angles
Because there are 2 exterior angles at each vertex, every triangle has a total of 6 exterior angles.
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4-2: Measuring Angles in Triangles

15. 4-2: Measuring Angles in Triangles
Remote Interior Angles:
A remote interior angle of a triangle is one of the 2 interior angles not adjacent to an exterior angle.
Remote interior angles change depending on which interior angle is being discussed.
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16. 4-2: Measuring Angles in Triangles
Remote Interior Angles:
∠ __ and ∠__ are remote
interior angles for exterior ∠BCD. B A C D
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17. 4-2: Measuring Angles in Triangles
Remote Interior Angles:
∠__ and ∠__ are remote interior
angles for exterior ∠BAE. B E A C
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18. 4-2: Measuring Angles in Triangles
Exterior Angle Theorem:
The measure of an exterior angle of a triangle is the sum of the measures of
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19. Exterior Angle Theorem: A Proof
Given: ΔABC with exterior angle
∠BCD
Prove: m∠BCD = m∠A + m∠B B A C D
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4-2: Measuring Angles in Triangles

20. Given: ΔABC with exterior angle ∠BCD Prove: m∠BCD = m∠A + m∠B
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21. 4-2: Measuring Angles in Triangles
What is the measure of ∠A?
124° A
52°
First hour start here 4/19/05
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22. 4-2: Measuring Angles in Triangles
Corollary
A corollary is a theorem that is easily proven by another theorem.
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4-2: Measuring Angles in Triangles

23. Acute Angles Corollary
The acute angles of a right triangle are ____________. A C B
m∠A + m∠B =
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24. Obtuse Angle Corollary
Right Angle Corollary
There can be at most ___ right angle in a triangle.
Obtuse Angle Corollary
There can be at most ___ obtuse angle in a triangle.
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25. 4-2: Measuring Angles in Triangles
In triangle ABC, m∠A is 16 more than m∠B and m∠C is 29 more than m∠B.
Determine the measure of each angle.
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26. 4-2: Measuring Angles in Triangles
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27. 4-2: Measuring Angles in Triangles
If AB is perpendicular to BC, determine the measures of all 8 numbered angles.
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28. 4-2: Measuring Angles in Triangles
Assignment:
pages ,
# (odds), 36, 39, (all)
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