Question

find the value of f that will allow you to prove that the triangles are congruent by the sas congruence theorem.

a
b
c
j
k
l
2f - 2
112°
3f - 11
40
5f + 7°
52

(it says)
◯ f = 10
◯ there is no value of f that will allow us to prove that these triangles are congruent.
◯ f = 21
◯ f = 22

Explanation:

Step1: Use the SAS Congruence Theorem

For two triangles to be congruent by SAS, two sides and the included angle of one triangle must be equal to two sides and the included angle of the other triangle.

Step2: Set up equations

Set \( 2f - 2=40 \) and \( 3f - 11 = 52 \).

For \( 2f - 2=40 \):

Add \( 2 \) to both sides: \( 2f=40 + 2=42 \).
Divide both sides by \( 2 \): \( f=\frac{42}{2}=21 \).

For \( 3f - 11 = 52 \):

Add \( 11 \) to both sides: \( 3f=52 + 11=63 \).
Divide both sides by \( 3 \): \( f=\frac{63}{3}=21 \).
Also, check the angle:
If \( f = 21 \), then \( 5f+7=5\times21 + 7=105 + 7=112^{\circ}\), which matches the included angle \( 112^{\circ}\) in the first triangle.

Answer:

\( f = 21 \)