6 Chapter Chapter 2 Ratio, Proportion, and Triangle Applications

Congruent and Similar Triangles Section 6.5 Congruent and Similar Triangles

Chapter 1 / Whole Numbers and Introduction to Algebra Congruent Triangles Two triangles are congruent when they have the same shape and the same size. Corresponding angles are equal, and corresponding sides are equal.

Angle-Side-Angle (ASA) If the measures of two angles of a triangle equal the measures of two angles of another triangle, and the lengths of the sides between each pair of angles are equal, the triangles are congruent. For example, these two triangles are congruent by Angle-Side-Angle.

Side-Side-Side (SSS) If the lengths of the three sides of a triangle equal the lengths of the corresponding sides of another triangle, the triangles are congruent. For example, these two triangles are congruent by Side-Side-Side.

Side-Angle-Side (SAS) If the lengths of two sides of a triangle equal the lengths of corresponding sides of another triangle, and the measures of the angles between each pair of sides are equal, the triangles are congruent. For example, these two triangles are congruent by Side-Angle-Side.

Example Determine whether triangle MNO is congruent to triangle RQS. Since the lengths of all three sides of triangle MNO equal the lengths of all three sides of triangle RQS, the triangles are congruent.

Example Determine whether triangle GHI is congruent to triangle JKL. The triangles are NOT congruent. The angle measures are not the same.

Chapter 1 / Whole Numbers and Introduction to Algebra Similar Triangles Similar triangles are found in art, engineering, architecture, biology, and chemistry. Two triangles are similar when they have the same shape but not necessarily the same size.

Similar Triangles In similar triangles, the measures of corresponding angles are equal and corresponding sides are in proportion. d = 6 a = 3 b = 5 e = 10 c = 8 f = 16 Side a corresponds to side d, side b corresponds to side e, and side c corresponds to side f.

Example Find the ratio of corresponding sides for the similar triangles QRS and XYZ.

Example Given that the triangles are similar, find the missing length x. Since the triangles are similar, corresponding sides are in proportion.

Example Tammy Shultz, a firefighter, needs to estimate the height of a burning building. She estimates the length of her shadow to be 8 feet long and the length of the building’s shadow to be 60 feet long. Find the approximate height of the building if she is 5 feet tall.

Example The height of the building is about 37.5 feet.