The Angle-Side-Angle (ASA) criterion for triangle congruence states that two triangles are congruent if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle.
To use the ASA criterion, we need two pairs of corresponding angles and the side between them to be congruent.
Given ∆ABC and ∆EFG, we would already have some facts provided in the diagram, but the specific relationships needed are:
1. One pair of corresponding angles are congruent.
2. Another pair of corresponding angles are congruent.
3. The side between these pairs of angles in the two triangles are congruent.
Let's examine the options to determine which additional statements are necessary.
**Options:**
i. measure of angle B = measure of angle F
ii. BC = FG
iii. measure of angle A = measure of angle E
iv. FG = 3
v. measure of angle of B = measure of angle E
**Analysis:**
- Option i states that angle B in ∆ABC is equal to angle F in ∆EFG. This gives us one pair of corresponding angles.
- Option ii states that side BC is equal to side FG. This gives us the side between angles B and C in ∆ABC and angles F and G in ∆EFG. If we consider angle B = angle F, then the side BC (or FG) is between these pairs of angles.
- Option iii states that angle A in ∆ABC is equal to angle E in ∆EFG. This gives us another pair of corresponding angles.
- Option iv states that FG is a specific length (3 units). This provides information about a side length but not necessarily paired with angles from the diagram.
- Option v states that angle B equals angle E. This provides a different pair of corresponding angles than options i and iii.
We need two pairs of angles and one included side for ASA:
1. From options i and iii, we get two pairs of angles: angle B = angle F and angle A = angle E.
2. One included side congruent statement is necessary, which is implied but not directly stated (likely given by the diagram).
From this analysis, the crucial pairs of angles are given by options i and iii. Therefore, the correct answer is:
D. i and iii only
In addition to the facts in the diagram, which other statements are necessary to prove that ∆ABC is congruent to ∆EFG by the ASA criterion?
Options:
i. measure of angle B = measure of angle F
ii. BC = FG
iii. measure of angle A = measure of angle E
iv. FG = 3
v. measure of angle of B = measure of angle E
Answers:
A. i only
B. i or iv only
C. iii or v only
D. i and iii only
1 answer