Triangle Theorems

Overview

Triangles are fundamental shapes in geometry with many important properties and relationships. The theorems discussed here focus on specific types of triangles and their unique characteristics, including isosceles triangles, right triangles, and similar triangles.

Isosceles Triangle Theorem

Statement: In an isosceles triangle, the angles opposite the equal sides are equal.
Converse: If two angles in a triangle are equal, the sides opposite them are also equal. Why It Works:

This theorem arises from the symmetrical nature of isosceles triangles. If two sides are equal, their corresponding angles must also be equal, maintaining balance in the shape.

The Line Joining Midpoints

Statement:

The line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

Why It Works:

This theorem is a consequence of parallel line properties and similar triangles formed by the mid-segment. It is also known as the Midpoint Theorem.

Ratios in a Right Triangle

Statement: The ratio of the lengths of the sides in a right triangle follows Pythagoras' theorem.

c^2 = a^2 + b^2

Where $c$ is the hypotenuse, and $a$ , $b$ are the other two sides.

Why It Works:

This relationship holds because the square of the longest side is equal to the sum of the squares of the other two sides, forming the basis for trigonometry.

Proportionality in Similar Triangles

Statement: Corresponding sides of similar triangles are proportional.

Why It Works:

In similar triangles, corresponding angles are equal, which ensures proportional scaling between all corresponding sides.

Converse of Similarity

Statement: If the corresponding sides of two triangles are proportional, the triangles are similar.

Why It Works:

This converse establishes that proportional side lengths imply the equality of corresponding angles, proving similarity.

Worked Examples

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Example 1: Isosceles Triangle

Problem: In an isosceles triangle $\triangle ABC$ , $AB = AC$ , and $\angle B = 50^\circ$ .

Find $\angle C$ and $\angle A$

Solution:

Step 1: By the Isosceles Triangle Theorem, .

\angle B = \angle C = 50^\circ

Step 2: Use the Angle Sum Theorem:

\angle A = 180^\circ - (\angle B + \angle C) = 180^\circ - (50^\circ + 50^\circ) = 80^\circ

Answer: $\angle A = :success[80^\circ], \, \angle B = :success[50^\circ], \, \angle C = :success[50^\circ]$

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Example 2: The Line Joining Midpoints

Problem: In $\triangle ABC$ , $D$ and $E$ are midpoints of $AB$ and $AC$ , respectively.

Prove that $DE$ is parallel to $BC$ and find its length if $BC=10$

Solution:

Step 1: By the Midpoint Theorem, $DE \parallel BC$

Step 2: The length of $DE$ is half the length of $BC$ :

DE = \frac{1}{2} \times BC = \frac{1}{2} \times 10 = 5

Answer: $DE \parallel BC$ , and $DE = :success[5]$

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Example 3: Proportionality in Similar Triangles

Problem: $\triangle ABC \sim \triangle DEF$ with $AB = 6, BC = 8, EF = 12$ .

Find $DE$ .

Solution:

Step 1: Use the proportionality of corresponding sides:

\frac{AB}{DE} = \frac{BC}{EF}

Substitute the known values:

\frac{6}{DE} = \frac{8}{12}

Step 2: Solve for $DE$ :

DE = \frac{6 \times 12}{8} = 9

Answer: $DE = :success[9]$

Summary

Isosceles Triangle Theorem: Equal sides have equal opposite angles, and vice versa.
Midpoint Theorem: The line joining midpoints of two sides is parallel to the third side and half its length.
Pythagoras' Theorem: Relates the sides of right triangles.
Similarity Theorems:
- Proportional sides indicate similarity.
- Similar triangles have proportional corresponding sides. These theorems are crucial for solving problems and proving relationships in triangles.

Triangle Theorems Simplified Revision Notes for Leaving Cert Mathematics

Triangle Theorems

Overview

Isosceles Triangle Theorem

The Line Joining Midpoints

Ratios in a Right Triangle

Proportionality in Similar Triangles

Converse of Similarity

Worked Examples

Example 1: Isosceles Triangle

Example 2: The Line Joining Midpoints

Example 3: Proportionality in Similar Triangles

Summary

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