1. Warm-Up X = 5 X = 11 QS = RS = QR = 25 LN = MN = 29
Find the value of x and the measures of the unknown sides.
X = 5
QS = RS = QR = 25
X = 11
LN = MN = 29

2. Warm Up No 3 + 4 = 7 < 8 Yes, the triangle is obtuse
Is it possible to form a triangle with the given lengths? 3, 4, 8
Determine if three segments that are 7, 14, and 16 units long can form a triangle. If so, classify the triangle as acute, right, or obtuse.
No = 7 < 8
Yes, the triangle is obtuse

3. Unit 4 Lesson 3
Triangle Theorems

4. Objectives I can recognize triangle theorems
I can discover and apply theorems about triangles

5. Isosceles Triangle Theorem

6. Example – Isosceles Triangles
Name a pair of unmarked congruent segments.
___
BC is opposite D and BD is opposite BCD, so BC  BD.
Answer: BC  BD

7. Example - Isosceles Triangles
Which statement correctly names two congruent angles?
A. PJM  PMJ
B. JMK  JKM
C. KJP  JKP
D. PML  PLK A B C D

8. Example – Isosceles Triangles
ALGEBRA Find the value of each variable.
mDFE = 60
4x – 8 = 60
4x = 68
x = 17
DF = FE
6y + 3 = 8y – 5
3 = 2y – 5
8 = 2y
4 = y

9. Name two congruent segments if 1  2.B. C. D. A B C D

10. Exterior Angle Theorem

11. Example – Exterior Angle Theorem
Find the value of x and then
find the measure of both
angles.
mLOW + mOWL = mFLW
x + 32 = 2x – 48
32 = x – 48
80 = x
Answer: So, mFLW = 2(80) – 48 or 112.
and mF0W = 80

12. Practice Find the measure of each missing angle m1 = 104 m2 = 76

13. Practice
Find the measure of each missing angle

14. Angle – Side Relationships

15. In other words…
You can list the angles and sides of a triangle from smallest to largest (or vice versa)
The smallest side is opposite the smallest angle
The longest side is opposite the largest angle

16. Example List the angles of ΔABC in order from smallest to largest.
Answer: C, A, B

17. Example
List the sides of ΔRST in order from shortest to longest.
A. RS, RT, ST
B. RT, RS, ST
C. ST, RS, RT
D. RS, ST, RT A B C D

18. Finally, Comparing TWO triangles
Inequalities in two triangles
Compare how the side lengths and angles are related
What effect does changing these measures have on triangles?

19. The Hinge Theorem

20. Example – The Hinge Theorem
Compare the measures AD and BD.
In ΔACD and ΔBCD, AC  BC, CD  CD, and ACD > BCD.
Answer: By the Hinge Theorem, mACD > mBCD, so AD > DB.

21. Example – The Hinge Theorem
Compare the measures ABD and BDC.
In ΔABD and ΔBCD, AB  CD, BD  BD, and AD > BC.
Answer: By the Converse of the Hinge Theorem, ABD > BDC.

22. Example A B C D B. Compare JKM and KML. A. mJKM > mKML
B. mJKM < mKML
C. mJKM = mKML
D. not enough information A B C D