4. match the postulate that proves congruence
sss asa sas aas hl

a. ______

b. ______

c. ______

d. ______

e. ______

Question

4. match the postulate that proves congruence
sss        asa        sas        aas        hl

a. ______

b. ______

c. ______

d. ______

e. ______

Accepted Answer

# Explanation for each part:
## Part a:
### Step1: Identify congruent parts
The two triangles share a common side, have one pair of congruent angles (the right - like angles) and one pair of congruent sides (marked). Wait, actually, looking at the diagram, we have two sides and the included angle? No, wait, in the first diagram (a), we have two angles and the included side? Wait, no, let's recall the postulates:
- SAS: Side - Angle - Side (two sides and the included angle)
- ASA: Angle - Side - Angle (two angles and the included side)
- SSS: Side - Side - Side (three sides)
- AAS: Angle - Angle - Side (two angles and a non - included side)
- HL: Hypotenuse - Leg (for right triangles)

In diagram a, we have two angles (the marked angles) and the included side (the common side). Wait, no, actually, the two triangles have two sides and the included angle? Wait, no, let's re - examine. The first triangle (a) has two angles (one right - looking and one marked) and the included side? Wait, maybe I made a mistake. Wait, the correct approach:

For diagram a: We have two sides and the included angle? No, wait, the two triangles share a side, have one angle (marked) and another angle (the right - angle - like). Wait, no, let's think again. The postulate for a: We have two angles and the included side? No, actually, the correct postulate for a is SAS? Wait, no, let's go step by step.

Wait, the first diagram (a) has two triangles with two sides and the included angle? No, maybe it's SAS. Wait, no, let's look at the other diagrams:

## Part b:
### Step1: Identify congruent parts
The diagram b shows two right triangles with a common hypotenuse (the line with the right angles), and two sides marked as congruent. Wait, the right triangles have a leg and the hypotenuse? No, the markings: the vertical sides are marked, the horizontal sides? Wait, no, the diagram b has two triangles with right angles, a common side (the hypotenuse), and two legs marked. Wait, HL (Hypotenuse - Leg) is for right triangles where the hypotenuse and one leg are congruent. But in this case, maybe it's SAS? No, wait, the two triangles have two sides and the included angle (the right angle). Wait, no, the correct postulate for b is HL? Wait, no, let's recall:

HL is 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, then the triangles are congruent.

In diagram b, the two triangles are right - angled, share the hypotenuse, and have one leg congruent (marked). So HL? Wait, no, maybe it's SAS. Wait, I think I messed up. Let's start over.

### Correct analysis:

- **Diagram a**:
  - We have two sides (marked) and the included angle (the angle between them, marked). So this is SAS (Side - Angle - Side). Wait, no, the angle is between the two sides. So SAS.

- **Diagram b**:
  - The two triangles are right - angled, have a common hypotenuse, and one leg congruent (marked). So HL (Hypotenuse - Leg).

- **Diagram c**:
  - We have two sides (marked) and the included angle (the angle at the top, marked). So SAS? Wait, no, the two triangles have two sides (marked) and the included angle (the top angle). Wait, no, maybe ASA? Wait, no, the top angle is common, and two sides are marked. Wait, SSS? No, two sides and the included angle: SAS. Wait, no, the correct postulate for c is SAS? Wait, no, let's see:

- **Diagram d**:
  - The diagram d is a rhombus (a quadrilateral with four equal sides) divided into two triangles. The two triangles share a side, and the other two sides are marked as equal. So three sides: SSS (Side - Side - Side), because the two triangles have three sides congruent (the common side, and the two marked sides).

- **Diagram e**:
  - The diagram e shows two triangles with two angles (marked) and a non - included side (the marked side). So AAS (Angle - Angle - Side), because we have two angles and a side that is not included between them.

Now, let's re - assign:

a. SAS: Because we have two sides and the included angle.

b. HL: Because it's two right triangles with hypotenuse (common side) and one leg congruent.

c. SAS: Wait, no, c has two sides (marked) and the included angle (the top angle). Wait, no, c has two sides and the included angle, so SAS.

d. SSS: Because the two triangles in the rhombus have three sides congruent (the common diagonal, and the two pairs of equal sides of the rhombus).

e. AAS: Because we have two angles (marked) and a non - included side (marked).

Wait, maybe I made a mistake in a. Let's re - check a:

In diagram a, we have two triangles with two angles (one at the bottom, one at the top) and the included side (the common side). So that's ASA (Angle - Side - Angle). Oh! I see, I messed up earlier. So:

a. ASA: Two angles and the included side.

b. HL: Right triangles, hypotenuse and leg.

c. SAS: Two sides and the included angle.

d. SSS: Three sides (the two triangles in the rhombus have three sides equal: the common diagonal, and the two pairs of equal sides of the rhombus).

e. AAS: Two angles and a non - included side.

Yes, that makes more sense.

## Step - by - Step for each:

### a.
- Step1: Identify the congruent parts. The two triangles have two angles (marked) and the included side (the common side).
- Step2: Recall the postulate for Angle - Side - Angle (ASA), which states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. So the postulate is ASA.

### b.
- Step1: Identify the congruent parts. The two triangles are right - angled, have a common hypotenuse, and one leg congruent (marked).
- Step2: Recall the HL (Hypotenuse - Leg) postulate, which states that 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. So the postulate is HL.

### c.
- Step1: Identify the congruent parts. The two triangles have two sides (marked) and the included angle (the top angle).
- Step2: Recall the SAS (Side - Angle - Side) postulate, which states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent. So the postulate is SAS.

### d.
- Step1: Identify the congruent parts. The two triangles in the rhombus have three sides congruent (the common diagonal, and the two pairs of equal sides of the rhombus).
- Step2: Recall the SSS (Side - Side - Side) postulate, which states that if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. So the postulate is SSS.

### e.
- Step1: Identify the congruent parts. The two triangles have two angles (marked) and a non - included side (marked).
- Step2: Recall the AAS (Angle - Angle - Side) postulate, which states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, the triangles are congruent. So the postulate is AAS.

# Answer:
a. ASA
b. HL
c. SAS
d. SSS
e. AAS

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QUESTION IMAGE

Question

Explanation:

Part a:

Step1: Identify congruent parts

Part b:

Step1: Identify congruent parts

Correct analysis:

Answer: