Warm-Up 1 If the lengths of two sides of an isosceles triangle are 3 and 7, what is the perimeter of the triangle?

Warm-Up 2 Use your straightedge to draw an acute angle. How could you use your compass to turn this angle into an isosceles triangle?

Definition: Review isosceles triangle A triangle is an isosceles triangle if and only if it has at least two congruent sides.

4.7 Use Isosceles and Equilateral Triangles Objectives: 1.To complete and use the Base Angles Theorem and its converse 2.To deduce the Equilateral Triangle Theorem from the Base Angles Theorem

Investigation 1 Use the following investigation to complete the Base Angles Theorem. Half of the class should work with isosceles triangles with acute vertex angles, and the other half should work with isosceles triangles with obtuse vertex angles.

Investigation 1 1.Draw an angle and label the vertex C. This will be your vertex angle. 2.Using point C as center, swing an arc that intersects both sides of <C.

Investigation 1 3.Label the points of intersection A and B. Construct side AB. You have constructed isosceles Δ ABC with base AB. 4.Use your protractor to measure the base angles of isosceles Δ ABC.

Investigation 1 Compare your results with the rest of the class. What relationship do you notice about the base angles of each isosceles triangle?

Base Angles Theorem: If two sides of a triangle are congruent, then the angles opposite them are congruent.

Example 1 Median 1.AC = AB Given 2.BM = CM Def. of Median 3.AM = AM Reflex 4. AMC = AMB SSS 5.<B = <C CPCTC

Example 2 Altitude Try this one in your notebook

Example 3 Find the value of x. 35

Example 4: SAT If AB = BC, what is y in terms of x ? y=2x-4

Investigation 2 Use the following to investigate the converse of the Base Angles Theorem.

Investigation 2 1.Draw a line segment AB. 2.Draw an acute angle at A with your protractor.

Investigation 2 3.Duplicate  A at point B with your protractor. Label the point of intersection C. 4.Use your compass or ruler to compare AC and BC.

Converse of the Base Angles Theorem: If two angles of a triangle are congruent, then the sides opposite them are congruent.

Proof of Converse This proof is similar to the one for The Base Angles Theorem, but you draw an altitude instead of a median. Of course, the other proof would have worked with an altitude… Altitude

Investigation 3 In this Investigation, we will use some more compass and straightedge wizardry to construct a perfect equilateral triangle.

Investigation 3 1.Draw AB.

Investigation 3 2.Using A as center and AB as the radius, draw a quarter-circle arc through B.

Investigation 3 3.Using B as center and AB as the radius, draw a quarter-circle arc through A.

Investigation 3 4.Label the point of intersection C.

Investigation 3 5.Draw AC and BC.

Investigation 3 Click button below to watch a construction video

Investigation 3 In Δ ABC, measure each angle with the protractor. What do you notice about the angle measurements? Use deductive reasoning to show that these measurements must be true based on the Base Angles Theorem.

Equilateral Triangle Theorem A triangle is equilateral if and only if it is equiangular.

Example 5 Find the values of x and y in the diagram. Answer in your notebook

Example 6: SAT Find the value of x. 30

Example 7: SAT In the figure shown, if O is the center of the circle, what is the value of x? 55

Example 8: SAT In the figure shown, if AB = AC and AC||BD, what is the value of x? 65

Assignment P : 1, 2, 10-22, 30, 32-34, 40, 47, 48, 52, 53 Challenge Problems