The Altitude Splits It
Isosceles triangle ABC with the altitude AD drawn to the base.
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Dropping an altitude creates two right-angled triangles we can analyze.
Use congruence to prove that angles opposite equal sides are equal, and that each angle of an equilateral triangle is 60°.
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Using congruence to discover a new fact about isosceles triangles.
So far, we have used different conditions like SSS, SAS, and RHS just to test if two triangles are congruent. But congruence is more than a test—it is a powerful tool for proving and discovering new properties about geometric figures.
Isosceles triangle ABC with the altitude AD drawn to the base.
Dropping an altitude creates two right-angled triangles we can analyze.
The RHS proof that the two base angles of an isosceles triangle are equal.
In with , show that , and then find their exact measurements when .
Deriving the angles of an equilateral triangle from the isosceles result.
An equilateral triangle ABC has all three sides equal. Find the measures of its angles.
Fill in the steps from equal sides to 60 degree angles.
In a triangle, angles opposite to sides are equal. In an equilateral triangle all three are equal, so all three are equal. Because the interior angles of a triangle sum to 180 degrees, 3 times the angle of an equilateral triangle = degrees. So each angle is degrees. Similarly, in an isosceles triangle with an apex angle of 80 degrees, each base angle is exactly degrees.
Four angle problems using equal sides and the 180 degree sum.
In where and the apex , what is the measure of the base angle ?
Triangles repeating in a pyramid, a dome, a rangoli and a bridge.
Congruent triangles appear everywhere, giving structure and repeating beauty to the objects around us.