Angle Theorems

Overview

Understanding angle theorems is essential for solving problems involving parallel lines, transversals, and triangles. These theorems provide foundational tools for proving relationships between angles in geometric figures. Below is a detailed explanation of each theorem:

Vertically Opposite Angles

What It States: When two straight lines intersect at a point, the angles opposite each other (vertically opposite angles) are always equal.
Why It Works: At the point of intersection, the adjacent angles are supplementary, meaning their measures add up to $180^\circ$ . Since each pair of vertically opposite angles shares the same adjacent angles, they must have equal measures.

Alternate Angles

What It States: Alternate angles are formed when a transversal crosses two lines. These angles lie on opposite sides of the transversal but are inside the two lines.
- If the two lines are parallel, the alternate angles are equal.
- Converse: If the alternate angles formed by a transversal are equal, the lines it intersects are parallel.
Why It Works: When the lines are parallel, the transversal creates equal angles due to the consistent spacing between the two lines. This is a fundamental property of parallel lines.

Angle Sum in a Triangle

What It States: The sum of the interior angles in any triangle is always $180^\circ$ .
Why It Works: This can be proven by drawing a line parallel to one side of the triangle through the opposite vertex. This line creates alternate interior angles with the other two sides of the triangle, and their measures add up to form a straight angle ( $180^\circ$ ).

Exterior Angle of a Triangle

What It States: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior opposite angles.
Why It Works: The exterior angle forms a straight line with one of the interior angles of the triangle. Using the angle sum property ( $180^\circ$ ), we can deduce that the exterior angle is the sum of the other two interior angles.

Worked Examples

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Example 1: Vertically Opposite Angles

Problem: Two lines intersect at point $O$ .

If one angle measures $45^\circ$ , find all other angles.

Solution:

The angle opposite $45^\circ$ is also $45^\circ$ (vertically opposite angles are equal).
The remaining two angles are supplementary to $45^\circ$ :

180^\circ - 45^\circ = 135^\circ

The two $135^\circ$ angles are also vertically opposite and equal.

Answer: The angles are $45^\circ, 135^\circ, 45^\circ, 135^\circ$

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Example 2: Alternate Angles

Problem: A transversal intersects two lines, creating angles of $70^\circ$ and $x$ (alternate angles).

If the lines are parallel, find $x$ .

Solution:

By the Alternate Angle Theorem:

x = 70^\circ

Answer: $x = 70^\circ$

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Example 3: Angle Sum in a Triangle

Problem: A triangle has angles $50^\circ$ and $60^\circ$ . Find the third angle.

Solution:

By the Angle Sum Theorem:

\text{Third angle} = 180^\circ - (50^\circ + 60^\circ) = 70^\circ

Answer: The third angle is $70^\circ$

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Example 4: Exterior Angle of a Triangle

Problem: In a triangle, one exterior angle measures $120^\circ$ . If one interior opposite angle is $50^\circ$ , find the other interior opposite angle.

Solution:

By the Exterior Angle Theorem:

120^\circ = 50^\circ + \text{Other Interior Angle}

\text{Other Interior Angle} = 120^\circ - 50^\circ = 70^\circ

Answer: The other interior angle is $70^\circ$

Summary

Vertically Opposite Angles: Equal when two lines intersect.
Alternate Angles: Equal if the lines are parallel; if alternate angles are equal, the lines are parallel.
Angle Sum in a Triangle: The sum of angles is always $180^\circ$
Exterior Angle of a Triangle: Equal to the sum of the two interior opposite angles.
These theorems are essential for understanding relationships between angles in geometric figures.

Angle Theorems Simplified Revision Notes for Leaving Cert Mathematics

Angle Theorems

Overview

Vertically Opposite Angles

Alternate Angles

Angle Sum in a Triangle

Exterior Angle of a Triangle

Worked Examples

Example 1: Vertically Opposite Angles

Example 2: Alternate Angles

Example 3: Angle Sum in a Triangle

Example 4: Exterior Angle of a Triangle

Summary

500K+ Students Use These Powerful Tools to Master Angle Theorems For their Leaving Cert Exams.

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