Angle – Angle Triangle Similarity. corresponding angles are congruent

Key Concepts

• Determine Whether Triangles Are Similar
• Solve problems involving Similar Triangles

6.10 Angle – Angle Triangle Similarity 

Angle: 

Angles are formed when two lines intersect at a point. 

Angle – Angle Triangle Similarity: 

In two triangles, if two pairs of corresponding angles are congruent, then the triangles are similar.  

(Note that if two pairs of corresponding angles are congruent, then it can be shown that all three pairs of corresponding angles are congruent, by the Angle Sum Theorem.) 

If ∠A≅∠D and ∠B≅∠E 

then the triangles ΔABC and ΔDEF are similar. 

Determine Whether Triangles Are Similar 

Example1: 

Determine whether triangles △RUV and △RST are similar. 

Sol: 

Redraw the diagram as two triangles △RUV and △RST. 

From the diagram we know that both ∠48° 

So ∠RUV ≅ ∠RST. 

∠R ≅ ∠R by the reflexive property of congruence. 

By AA similarity △RUV ~ △RST.  

Example2: 

Explain whether the triangles △PQR and △STU are similar. 

Sol: 

Triangle PQR:  

Write the Triangle Sum Theorem for this triangle. 

maps + mm + m∠R = 180° 

Substitute the given angle measures. 

45° + 100° + m∠R = 180° 

m∠R = 35° 

Triangle STU:  

Write the Triangle Sum Theorem for this triangle. 

m∠S + m∠T + m∠U = 180° 

Substitute the given angle measures. 

m∠S + 100° + 35° = 180° 

m∠S = 45° 

Conclusion: 

Three Angles of triangle PQR are 45°, 100° and 35°. 

Three Angles of triangle STU are 45°, 100° and 35°. 

Because two angles in one triangle are congruent to two angles in the other triangle, the two triangles are similar. 

Solve problems involving Similar Triangles 

Example: 

In the figure Δ ABC ~ Δ EDC. Solve for x.  

Sol: 

Condition of similar triangles: 

If two corresponding angles are equal. 

If three corresponding sides are in same ratio. 

The ratio of two pairs of corresponding sides are equal and their included angle are equal. 

As we are given that: 

So, 

∠B = ∠D 

(8x + 16) ° =120° 

8x=120°- 16° 

8x = 104° 

x= 104/8

x=13 

Therefore, the value of ‘x’ is, 13 

Exercise

1. Determine whether the two triangles given below are similar. Justify your answer

2. Determine whether the two triangles given below are similar. Justify your answer.

3. Determine whether the two triangles given below are similar.

4. Determine whether the two triangles given below are similar. Justify your answer.

5. Are the two triangles are similar? Explain.

6. Are the two triangles are similar? Explain.

7. What does it mean for two shapes to be similar?
   • Same shape, but different size.
   • Same size and same shape.
   • Their angles add up to 180o.
   • They are congruent.

8. What is angle-angle triangle similarity?     
9. Are triangle RST and NSP similar? Explain.

9. Find the value of x and y given that ABC is similar to MNC.

Concept Map: 

What have we learned:

• Understand Angle – Angle Triangle Similarity
• Determine Whether Triangles Are Similar
• Solve problems involving Similar Triangles