AA Similarity

Show your work for each problem.

1.
What does the AA in AA Similarity stand for?
1. a)Angle-Angle
2. b)Altitude-Altitude
3. c)Adjacent-Angle
4. d)Area-Area
Answer: Angle-Angle
AA stands for Angle-Angle. If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
2.
Triangle ABC has and . Triangle DEF has and . Are the triangles similar?
1. a)No, the angles are different
2. b)Yes, by SSS Similarity
3. c)Cannot be determined
4. d)Yes, by AA Similarity
Answer: Yes, by AA Similarity
Since and , the triangles have two pairs of congruent angles. By AA Similarity, .
3.
In a triangle, one angle is and another is . What is the third angle?
Answer: 50
The third angle = . This is why AA Similarity works: if two angles are equal, the third must be equal too!
4.
Why do we only need two pairs of congruent angles to prove triangles similar?
1. a)Because the third angle is determined by the sum
2. b)Because the third angle is always
3. c)Because triangles only have two angles
4. d)Because sides don't matter
Answer: Because the third angle is determined by the sum
Since all triangle angles sum to , if two angles are equal, the third angle is automatically equal: gives the same result for both triangles.
5.
If and , what is ?
Answer: 35
In the similarity statement , the first letters correspond: A↔D, B↔E, C↔F. So .
6.
Triangle PQR has and . Triangle XYZ has and . Are the triangles similar?
Answer: Yes
1. Find in triangle PQR: 65
2. Find in triangle XYZ: 67
3. List PQR angles: . List XYZ angles: . Do all three angles match? (yes/no) yes
4. Are the triangles similar? (yes/no) yes
7.
If line DE is parallel to line BC in triangle ABC, which triangles are similar?
1. a)No triangles are similar
2. b)
3. c)
4. d)
Answer:
When DE ∥ BC, angle A is shared, and (corresponding angles with parallel lines). By AA Similarity, .
8.
A 2m tall person casts a 3m shadow. A tree casts a 15m shadow at the same time. Using AA similarity, how tall is the tree (in meters)?
Answer: 10
. Cross multiply: , so , giving meters.
9.
Triangle ABC has and . Triangle DEF has and . Write the correct similarity statement.
Answer: ABC ~ DEF
1. Find in triangle ABC: 55
2. Find in triangle DEF: 90
3. Match the angles: . Which angle in DEF equals ? E
4. So A↔D, B↔E, C↔F. Write: ___ DEF
10.
Does AA Similarity prove that triangles are congruent?
1. a)Yes, if all three angles match
2. b)Only if the triangles are right triangles
3. c)Yes, same angles means same triangle
4. d)No, only similarity (same shape, not same size)
Answer: No, only similarity (same shape, not same size)
AA proves similarity, not congruence. Similar triangles have the same shape (angles) but can be different sizes. To prove congruence, you also need information about side lengths.
11.
In triangle ABC, point D is on AB and point E is on AC such that DE ∥ BC. If and , find and .
Answer: 50, 65
1. Since DE ∥ BC, what is the relationship between and ? equal
2. Therefore, ° 50
3. What is the relationship between and ? equal
4. Therefore, ° 65
12.
Two similar triangles have sides in the ratio . If the smaller triangle has a perimeter of 24 cm, what is the perimeter of the larger triangle (in cm)?
Answer: 40
If sides are in ratio , perimeters are also in ratio . So . Cross multiply: , giving cm.
13.
A lamppost 6m tall casts a shadow of 4m. A person standing nearby casts a shadow of 1.2m at the same time. How tall is the person (in meters)?
Answer: 1.8
1. Why are the triangles formed by the objects and shadows similar? same sun angle
2. Set up the proportion: . Write as: 1.2
3. Simplify to lowest terms 3/2
4. Solve: . Cross multiply to get 1.8
14.
If with a scale factor of , and the area of is 16 square units, what is the area of ?
1. a)32 square units
2. b)24 square units
3. c)48 square units
4. d)36 square units
Answer: 36 square units
If sides are in ratio , areas are in ratio . So , giving square units.
15.
In triangle ABC, . Point D is on BC such that AD ⊥ BC. If , find (in degrees).
Answer: 30
In right triangle ABC: . In right triangle ADC, and , so .