Question

If \(\triangle P R Q \cong \triangle Y X Z, m \angle P=63, m \angle Q=57, X Y=10\), and \(Y Z=11\), find each measure. \(m \angle R\)

Short answer

The measure of \( \angle R \) is \( 60^\circ \).

Step-by-step solution

Understand Triangle Congruence

Since \( \triangle PRQ \cong \triangle YXZ \), it means that all corresponding sides and angles of the triangles are equal.

Use Angle Sum Property

The sum of angles in any triangle is \( 180^\circ \). So for \( \triangle PRQ \), we have \( m\angle P + m\angle Q + m\angle R = 180 \).

Substitute Known Values

Substitute the given values into the equation: \( 63 + 57 + m\angle R = 180 \).

Solve for \( m\angle R \)

Calculate \( m\angle R \) by performing the subtraction: \( 120 + m\angle R = 180 \). Simplify this to find \( m\angle R = 180 - 120 = 60 \).

Conclusion

Therefore, the measure of \( \angle R \) is \( 60^\circ \).

Key concepts

Angle Sum Property

The Angle Sum Property is a fundamental concept in geometry, especially when dealing with triangles. This property states that the sum of all interior angles of a triangle is always equal to 180 degrees. In simple terms, if you add up the three angles inside any triangle, they will always total 180 degrees.
To use this property effectively, you can set up an equation. For example, if you have a triangle labeled as \(\triangle ABC\), you can represent the angles as \(m\angle A, m\angle B,\) and \(m\angle C\). The sum would then be \(m\angle A + m\angle B + m\angle C = 180\).
Knowing two angles can help you easily calculate the third angle using this property. This concept is not only essential for solving problems involving triangle measures, but it also lays a foundation for understanding more comprehensive geometric proofs and theorems.

• Helps to calculate unknown angles
• Applies universally to all triangles

Triangles

Triangles are one of the most basic shapes in geometry. They consist of three sides and three angles, making them a polygon. There are several types of triangles based on their sides and angles.
Based on side lengths, triangles can be:

• Equilateral: All sides and angles are equal. Each angle measures 60 degrees.
• Isosceles: Two sides (and consequently two angles) are equal.
• Scalene: All sides (and all angles) differ.

Based on angles, triangles can be categorized as:

• Acute: All angles are less than 90 degrees.
• Right: One angle is exactly 90 degrees.
• Obtuse: One angle is greater than 90 degrees.

Triangles are essential not just in mathematics but also in various fields such as engineering, architecture, and art. Their properties are used to calculate distances, forces, and in trigonometric functions, making them crucial in both practical and theoretical contexts.

Congruent Triangles

Congruent triangles are triangles that are identical in shape and size, although their orientation may differ. This means all their corresponding sides and angles are equal.
Two triangles are congruent if, and only if, they satisfy any one of the congruence postulates simultaneously, such as:

• SSS (Side-Side-Side): All three pairs of corresponding sides are equal.
• SAS (Side-Angle-Side): Two pairs of corresponding sides and the angle between them are equal.
• ASA (Angle-Side-Angle): Two pairs of corresponding angles and a side between them are equal.
• AAS (Angle-Angle-Side): Two pairs of corresponding angles and the side that is not between them are equal.
• HL (Hypotenuse-Leg): Used specifically for right triangles, where the hypotenuse and one leg are equal in both triangles.

Congruent triangles are crucial in proving various geometric theorems and calculating dimensions indirectly. Knowing triangles are congruent enables you to derive properties of one from the other effortlessly, such as knowing the measures of unknown angles or sides. This is particularly beneficial in geometry problems where direct measurement is not possible.