Question

2-114. which pairs of triangles below are congruent and/or similar? for each part, explain how you know using an appropriate triangle congruence or similarity condition. note: the diagrams are not necessarily drawn to scale. homework help
2-117. solve for the missing lengths in the sets of similar figures below. homework help
a. abcd ~ jklm

Explanation:

Step1: Recall triangle congruence and similarity conditions

For congruence: SSS (Side - Side - Side), SAS (Side - Angle - Side), ASA (Angle - Side - Angle), AAS (Angle - Angle - Side), HL (Hypotenuse - Leg for right - triangles). For similarity: AA (Angle - Angle), SSS (proportional sides), SAS (proportional sides and included angle).

Step2: Analyze part a

Check side - side ratios. $\frac{6}{2}=3$, $\frac{12}{4} = 3$, $\frac{15}{5}=3$. Since the ratios of corresponding sides are equal, the triangles are similar by SSS similarity.

Step3: Analyze part b

Find the third angle in each triangle. In the first triangle, the third angle is $180^{\circ}-(112^{\circ}+20^{\circ}) = 48^{\circ}$. In the second triangle, the third angle is $180^{\circ}-(45^{\circ}+20^{\circ})=115^{\circ}$. Since the angles are not equal, the triangles are neither congruent nor similar.

Step4: Analyze part c

We have two right - triangles. The first right - triangle has sides 12 m and 13 m (hypotenuse), and the second has sides 5 m and 12 m. By the HL (Hypotenuse - Leg) congruence criterion for right - triangles (if the hypotenuse and one leg of a right - triangle are congruent to the hypotenuse and one leg of another right - triangle, the triangles are congruent), the two right - triangles are congruent.

Step5: Analyze part d

The two triangles have two pairs of equal angles ($48^{\circ}$ and $62^{\circ}$). By the AA (Angle - Angle) similarity criterion, the triangles are similar.

Answer:

a. Similar (by SSS similarity)
b. Neither congruent nor similar
c. Congruent (by HL congruence for right - triangles)
d. Similar (by AA similarity)