Question

If \(\triangle P R Q \cong \triangle Y X Z, m \angle P=63\), and \(m \angle Q=57\), find \(m \angle X\).

Short answer

\( m \angle X = 60 \) degrees.

Step-by-step solution

Understand Congruency

Triangles \( riangle PRQ \) and \( riangle YXZ \) are congruent, which means all corresponding sides and angles are equal. Therefore, \( m \angle P = m \angle Y \), \( m \angle R = m \angle X \), and \( m \angle Q = m \angle Z \).

Find the Unknown Angle in \( riangle PRQ \)

For any triangle, the sum of the internal angles is always 180 degrees. We know \( m \angle P = 63\) and \( m \angle Q = 57 \). Find \( m \angle R \) using the equation: \[ m \angle R = 180 - (m \angle P + m \angle Q) = 180 - (63 + 57) \] Simplifying, we get \[ m \angle R = 180 - 120 = 60 \text{ degrees} \].

Determine the Measure of \( \\angle X \)

Since \( riangle PRQ \cong riangle YXZ \), \( m \angle X = m \angle R = 60 \) degrees by the property of congruent triangles having corresponding angles with equal measures.

Key concepts

Angle Measures in Triangles

Understanding angle measures in triangles is crucial for solving many geometry problems. A triangle is a three-sided polygon with three angles inside it. The angle measures refer to the sizes of these angles, usually given in degrees. To find the measure of any angle in a triangle, we use information about the other angles. This involves calculations based on known angles or the use of certain angle relationships.

Knowing just two angles in a triangle can help us find the third one. If you come across unusual situations involving special triangles, such as isosceles or equilateral triangles, remember that specific rules apply. For instance, in an isosceles triangle, two angles are equal. In an equilateral triangle, all angles measure 60 degrees each.

Here's a handy tip: When working with angle measures, expressing them clearly and noting their relationships can simplify problems. Always list what you know before diving into calculations.

Properties of Congruent Triangles

Congruent triangles are triangles that are identical in shape and size. This means all their corresponding sides have the same length, and their corresponding angles are equal. Thus, if two triangles are congruent, knowing the measurements of one can help you determine the measurements of the other.

When you see the symbol \(\triangle PQR \cong \triangle YXZ\), it indicates congruency. Some properties of congruent triangles are:

• Corresponding sides are equal: If \( \triangle PQR \cong \triangle YXZ \), then \( PQ = YX \), \( PR = YZ \), and \( QR = XZ \).
• Corresponding angles are equal: \( m\angle P = m\angle Y \), \( m\angle Q = m\angle Z \), and \( m\angle R = m\angle X \).

Knowing these properties can drastically simplify problems. Once you establish two triangles are congruent, transferring known angles or side lengths from one triangle to another becomes straightforward. Remember, congruency is a powerful tool in geometry that helps maintain the equivalence between figures.

Sum of Angles in a Triangle

A fundamental principle in geometry is that the sum of angles in any triangle is always 180 degrees. This rule applies to every type of triangle, whether they are scalene, isosceles, or equilateral.

Using this principle helps solve for unknown angles easily. For example, if you know two angles in a triangle, subtract their sum from 180 to find the third angle. Let's break it down:

• Write down the known angles: \( m\angle A \) and \( m\angle B \).
• Apply the angle sum rule: \( m\angle C = 180 - (m\angle A + m\angle B) \).

In exercises involving congruent triangles, this rule becomes even more powerful. Once you know a corresponding angle from a congruent triangle, you can substitute it directly and simplify your work. Mastering this concept is key to tackling a wide range of geometric challenges.