Question

determine if the two triangles are congruent. if they are, state how you know.
9)
10)
11)
12)
13)
14)
15)
16)
find the value of x.
17)
18)
19)
20)

Explanation:

Step1: Recall the triangle congruence criteria

The main congruence criteria for triangles are SSS (Side - Side - Side), SAS (Side - Angle - Side), ASA (Angle - Side - Angle), AAS (Angle - Angle - Side), and HL (Hypotenuse - Leg) for right - angled triangles.

Step2: Analyze problem 9

In problem 9, we have two angles and a non - included side. By the AAS (Angle - Angle - Side) congruence criterion, the two triangles are congruent.

Step3: Analyze problem 10

In problem 10, we have two angles and a non - included side. By the AAS (Angle - Angle - Side) congruence criterion, the two triangles are congruent.

Step4: Analyze problem 11

In problem 11, we have two sides and the included angle. By the SAS (Side - Angle - Side) congruence criterion, the two triangles are congruent.

Step5: Analyze problem 12

In problem 12, we have two angles and the included side. By the ASA (Angle - Side - Angle) congruence criterion, the two triangles are congruent.

Step6: Analyze problem 13

In problem 13, we have two sides and the included angle. By the SAS (Side - Angle - Side) congruence criterion, the two triangles are congruent.

Step7: Analyze problem 14

In problem 14, we have three sides. By the SSS (Side - Side - Side) congruence criterion, the two triangles are congruent.

Step8: Analyze problem 15

In problem 15, we have two angles and the included side. By the ASA (Angle - Side - Angle) congruence criterion, the two triangles are congruent.

Step9: Analyze problem 16

In problem 16, we have three sides. By the SSS (Side - Side - Side) congruence criterion, the two triangles are congruent.

Step10: Analyze problem 17

In problem 17, since the triangle has two equal sides (isosceles triangle), the base angles are equal. Using the angle - sum property of a triangle (\(x + 71+71 = 180\)), we solve for \(x\).

$$x=180-(71 + 71)=38^{\circ}$$

Step11: Analyze problem 18

In problem 18, since the triangle has two equal sides (isosceles triangle), the base angles are equal. Using the angle - sum property of a triangle (\(x + 75+75 = 180\)), we solve for \(x\).

$$x=180-(75 + 75)=30^{\circ}$$

Step12: Analyze problem 19

In problem 19, since the triangle has two equal sides (isosceles triangle), the base angles are equal. Using the angle - sum property of a triangle (\(x + 50+50 = 180\)), we solve for \(x\).

$$x=180-(50 + 50)=80^{\circ}$$

Step13: Analyze problem 20

In problem 20, since the triangle has two equal sides (isosceles triangle), the base angles are equal. Using the angle - sum property of a triangle (\(x + 65+65 = 180\)), we solve for \(x\).

$$x=180-(65 + 65)=50^{\circ}$$

Answer:

1. Congruent by AAS.
2. Congruent by AAS.
3. Congruent by SAS.
4. Congruent by ASA.
5. Congruent by SAS.
6. Congruent by SSS.
7. Congruent by ASA.
8. Congruent by SSS.
9. \(x = 38^{\circ}\)
10. \(x = 30^{\circ}\)
11. \(x = 80^{\circ}\)
12. \(x = 50^{\circ}\)