Math Problem Statement

Find all unknown angle measures in the pair of similar triangles. The known angles in one triangle are ∠B = 65° and ∠C = 55°, and the triangles are similar.

Solution

The image shows two triangles, and you're asked to find the unknown angle measures given that the triangles are similar. The known angles are:

  • In triangle ABC, B=65\angle B = 65^\circ and C=55\angle C = 55^\circ.
  • The second triangle is triangle MNP, which is similar to triangle ABC, meaning corresponding angles are equal.

Step 1: Calculate the missing angle in triangle ABC

In any triangle, the sum of the angles is 180180^\circ. Therefore, to find A\angle A: A=1806555=60.\angle A = 180^\circ - 65^\circ - 55^\circ = 60^\circ. So, A=60\angle A = 60^\circ.

Step 2: Corresponding angles in similar triangles

Since the triangles are similar, the corresponding angles are congruent:

  • AM\angle A \leftrightarrow \angle M,
  • BN\angle B \leftrightarrow \angle N,
  • CP\angle C \leftrightarrow \angle P.

This gives us:

  • mM=mA=60m \angle M = m \angle A = 60^\circ,
  • mN=mB=65m \angle N = m \angle B = 65^\circ,
  • mP=mC=55m \angle P = m \angle C = 55^\circ.

Final answers:

  • mA=60m \angle A = 60^\circ,
  • mM=60m \angle M = 60^\circ,
  • mN=65m \angle N = 65^\circ,
  • mP=55m \angle P = 55^\circ.

Would you like further details or have any other questions?

Here are 5 related questions to deepen your understanding:

  1. What are the properties of similar triangles?
  2. How can you prove that two triangles are similar?
  3. How do you use angle sums in polygons with more than three sides?
  4. How does the proportionality of sides relate to the angles in similar triangles?
  5. Can two triangles be similar if they share only one corresponding angle?

Tip: Always double-check the sum of angles in a triangle to ensure your calculations are correct!

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Sum Theorem
Similar Triangles

Formulas

Sum of interior angles in a triangle is 180°

Theorems

Triangle Sum Theorem
Similarity of Triangles

Suitable Grade Level

Grades 7-10