9.4 The Geometry of Triangles:
Congruence, Similarity, and the
Pythagorean Theorem
Part 2: Similar Triangles and the
Pythagorean Theorem
Similar Triangles
• Triangles that are exactly the same shape but not necessarily
the same size are similar triangles.
• For ΔABC to be similar to ΔDEF, the following must be true:
1. Corresponding angles must have the same measure.
2. The ratios of the corresponding sides must be the same
(in other words, corresponding sides are proportional).
Angle-Angle (AA) Similarity: If the measure of two angles of
one triangle are equal to those of tow corresponding angles
of a second triangle, then the two triangles are similar.
Verifying Similarity
• AB is parallel to ED. How can we verify that
ΔABC is similar to ΔEDC?
Finding Side Lengths and Angle
Measures in Similar Triangles
• ΔEDF is similar to ΔCAB with A = 92 and E = 46.
Find the unknown parts of each triangle.
Finding the Height of a Flagpole
• Lucie wants to measure the height of a flagpole. She notices
that at the time the shadow of the building is 18 feet long, the
shadow of the flagpole is 99 feet long. If the building is 10
feet high, what is the height of the flagpole?
Find the unknown parts of the
similar triangle STU and ZXY.
The Pythagorean Theorem
• Recall that in a right triangle, the side opposite the right angle
(also the longest side) is called the hypotenuse.
• The other two sides, which are
perpendicular, are called the legs.
Pythagorean Theorem: If the two legs of a
right triangle have lengths a and b, and the
hypotenuse has length c, then a2 + b2 = c2.
(In other words, the sum of the squares of the lengths of
the legs is equal to the square of the hypotenuse.)
• Natural numbers such as 3, 4, and 5 which satisfy the
Pythagorean Theorem are called a Pythagorean triple.
Using the Pythagorean Theorem
• Find the length of a.
A 10-foot ladder has its base 6
feet from a wall. How high up the
wall does the ladder reach?