The document discusses the hinge theorem and its converse in geometry. The hinge theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle. The document provides examples applying the hinge theorem and its converse to various triangle relationships. It asks students to compare sides and angles of triangles based on given congruent sides and angle measures.

1. Hinge Theorem
JezelynC.Fabelinia

2. ACTIVITY TIME!!!
Group 1 Group 2
Group 3

3. Guide Questions:
Guide questions:
1.What is your object?
2. What do you observe in the given object?
3. What do you observe when the object is
moving?

4. Hinge Theorem
𝑅
𝑆
𝑇
𝑀
𝑁
𝑂
Point R and N are considered the Hinge
of pair of non- congruent sides

5. Hinge Theorem
If two sides of a triangle are
congruent to two sides of
another triangle and the
included angle of the first is
larger than the included angle of
the second, then the third side
of the first triangle is longer
than the third side of the second
triangle.
∆𝑅𝑆𝑇
∆𝑁𝑀𝑂
𝑅𝑆 ≅ 𝑁𝑀
𝑅𝑇 ≅ 𝑁0
∠𝑅 ≅ ∠𝑁
m∠𝑅 > ∠𝑁
𝑅
𝑆
𝑇
𝑁
𝑀
𝑂
80 50°

6. Hinge Theorem

7. Example:
GIVEN:
RT≅ 𝑈𝑊
RS≅ 𝑈𝑉
m∠SRT=54°
m∠VUW=52°
ST 𝑉𝑊
>

8. Example:
GIVEN:
BC≅ 𝐸𝐶
AC≅ 𝐷𝐶
mAB=29𝑐𝑚
mDE=31𝑐𝑚
m∠1 m∠2
<
A
B
C
D
E

9. Example:
GIVEN:
AB≅ 𝑋𝑌
AC≅ 𝑋𝑌
m∠ABC=30°
m∠XYZ=60°
BC YZ
<

10. Example:
GIVEN:
BC≅ 𝐵𝐷
BA≅ 𝐵𝐴
m∠ABC=(3𝑋 − 9)°
m∠ABD=36°
AC AD
<

11. Example:
GIVEN:
RU≅ 𝑇𝑈
US≅ 𝑆𝑈
m∠RUS=60°
m∠TUS=(5X−20)°
RS TS
>

12. Hinge Converse Theorem
If two sides of a triangle are congruent to two sides of another
triangle and the third side of the first is longer than the third side
of the second, then the included angle in the first triangle is
greater than the included angle in the second triangle.
If we return to the alligator analogy, the converse of the
Hinge Theorem would tell us that the wider the alligator
opens his mouth (EF > BC), the larger the angle he creates at
the hinge of his jaw
(m∠D > m∠B). If EF > BC, then m∠D > m∠B.

13. Hinge Theorem
If two sides of a triangle are
congruent to two sides of
another triangle and the
included angle of the first is
larger than the included angle of
the second, then the third side
of the first triangle is longer
than the third side of the second
triangle.
∆𝑅𝑆𝑇
∆𝑁𝑀𝑂
𝑅𝑆 ≅ 𝑁𝑀
𝑅𝑇 ≅ 𝑁0
ST ≅ MO
𝑅
𝑆
𝑇
𝑁
𝑀
𝑂
ST > MO

14. <
EXAMPLE: Fill the box with >, <, or =.
GIVEN:
𝐴𝐶 ≅ 𝐵𝐶
m∠ACR=18°
m∠BCR=32°
𝐴𝑅 B𝑅

15. <
EXAMPLE: Fill the box with >, <, or =.
GIVEN:
𝐴𝐵 ≅ 𝑋𝑌
m∠ABC=30°
m∠XYZ=60°
B𝐶 YZ

16. >
EXAMPLE: Fill the box with >, <, or =.
GIVEN:
𝐴𝐵 ≅ 𝐸𝐷
m∠ABC=59°
m∠DEF=54°
A𝐶 EF

17. <
EXAMPLE: Fill the box with >, <, or =.
GIVEN:
LY≅ 𝐿𝐼
LN≅ 𝐿𝑁
m∠LYN=45°
m∠LIN=115°
𝑌𝑁 IN
L
N
Y I

18. >
EXAMPLE: Fill the box with >, <, or =.
GIVEN:
RS≅ 𝑅𝑈
RT≅ 𝑅𝑇
m∠RST=45°
m∠RUT=41°
𝑆𝑇 U
T

19. SEATWORK: Fill the box with >, <, or =.
1. 𝑀𝑅__𝑃𝑅 , ∠NMR__∠NPR 2. ∠ABC__∠DEF, AB__𝐷𝐸
3. 𝑀𝑃 __𝑅𝑄, ∠N ___∠S 4. ∠ADB ___∠CDB, 𝐴𝐵___𝐶𝐵