Congruent Triangles

Show your work for each problem.

1.
Two triangles are congruent. What can you conclude?
1. a)They have the same shape but different sizes
2. b)They have the same shape and size
3. c)They have neither the same shape nor size
4. d)They have different shapes but the same size
Answer: They have the same shape and size
Congruent triangles have the same shape AND the same size. All corresponding sides are equal, and all corresponding angles are equal.
2.
Which criterion uses three pairs of equal sides to prove congruence?
1. a)SAS
2. b)ASA
3. c)AAS
4. d)SSS
Answer: SSS
SSS (Side-Side-Side) uses three pairs of equal sides. If all three sides of one triangle equal the three sides of another, the triangles are congruent.
3.
For SAS congruence, where must the angle be located?
1. a)Opposite to the longer side
2. b)Outside the triangle
3. c)Between the two sides
4. d)At any vertex
Answer: Between the two sides
For SAS (Side-Angle-Side), the angle must be the included angle - the angle formed between the two given sides. This is crucial because having the angle elsewhere gives SSA, which doesn't prove congruence.
4.
If and cm, what is the length of in cm?
Answer: 8
Since , the corresponding sides are equal. Side corresponds to side , so cm.
5.
In congruent triangles, . If corresponds to , what is in degrees?
Answer: 65
In congruent triangles, corresponding angles are equal. Since corresponds to , we have .
6.
Which is NOT a valid criterion for proving triangle congruence?
1. a)SSS (Side-Side-Side)
2. b)AAA (Angle-Angle-Angle)
3. c)SAS (Side-Angle-Side)
4. d)ASA (Angle-Side-Angle)
Answer: AAA (Angle-Angle-Angle)
AAA only proves that triangles are similar (same shape), not congruent. Two triangles can have identical angles but different sizes - think of a small triangle and an enlarged copy of it.
7.
Triangles have: cm, cm, and . Can you prove they are congruent?
1. a)Yes, by ASA
2. b)Yes, by SAS
3. c)Yes, by SSS
4. d)No, this is SSA which doesn't prove congruence
Answer: No, this is SSA which doesn't prove congruence
The angle C is at vertex C, which is not between sides AB and BC (angle B would be between them). This is SSA (Side-Side-Angle), which does NOT prove congruence due to the ambiguous case.
8.
Two triangles are congruent by SSS. The sides of the first triangle are 6 cm, 8 cm, and 10 cm. What is the perimeter of the second triangle in cm?
Answer: 24
Since the triangles are congruent, the second triangle has the same side lengths: 6 cm, 8 cm, and 10 cm. Perimeter cm.
9.
Given: cm, , . Which criterion proves ?
Answer: ASA
1. How many equal sides are given? 1
2. How many equal angles are given? 2
3. Is the side AB between angles A and B? yes
4. What criterion has Angle-Side-Angle? (Enter the abbreviation) ASA
10.
Right triangles ABC and DEF have right angles at B and E. If cm and cm, prove they are congruent.
Answer: HL
1. What type of triangles are ABC and DEF? right triangles
2. In a right triangle, what is the side opposite the right angle called? hypotenuse
3. Which sides are the hypotenuses? (AC or AB for triangle ABC) AC
4. What is AB in triangle ABC? leg
5. Which criterion uses hypotenuse and leg for right triangles? HL
11.
In , and . If , what is in degrees?
Answer: 50
First, . Since , corresponding angles are equal: .
12.
Why is HL (Hypotenuse-Leg) a valid criterion but SSA (Side-Side-Angle) is not?
1. a)HL requires a 90-degree angle which eliminates the ambiguous case
2. b)SSA can only be used for acute triangles
3. c)HL uses more information than SSA
4. d)HL was invented after SSA was disproven
Answer: HL requires a 90-degree angle which eliminates the ambiguous case
SSA fails because the same two sides and non-included angle can form two different triangles. However, HL includes a 90-degree angle, which eliminates this ambiguity - the Pythagorean theorem guarantees only one possible triangle.
13.
In quadrilateral ABCD, diagonal AC divides it into triangles ABC and ACD. If , , and is shared, prove .
Answer: SSS
1. In triangle ABC, list the three sides AB, BC, AC
2. In triangle CDA, list the three sides CD, DA, AC
3. Which side of ABC equals which side of CDA? (AB equals...) CD
4. Which side of ABC equals which side of CDA? (BC equals...) DA
5. What about AC in both triangles? shared
6. All three pairs of sides are equal. Which criterion is this? SSS
14.
Two congruent triangles have areas of 24 cm each. If one triangle has a base of 8 cm, what is the height of the other triangle to a corresponding base in cm?
Answer: 6
Area . So , giving cm. Since triangles are congruent, corresponding heights are equal, so the other triangle also has height 6 cm.
15.
Given: , , cm. Prove and find the criterion.
Answer: ASA
1. How many pairs of equal angles do we have? 2
2. Is the side PQ between angles P and Q? yes
3. Is the side ST between angles S and T? yes
4. We have two angles and the included side. What criterion is this? ASA