# The isosceles triangle theorem and its converse may

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The Isosceles Triangle Theorem and its converse may be used to prove the Equilateral Triangle Theorem and its converse. This is done in Activity 2.3.6 Equilateral Triangles . In Activity 2.3.7 Altitudes and Medians students use coordinate geometry to show that in scalene triangles the altitude and median are distinct line segments, but in isosceles triangles the altitude and median drawn from the vertex angle to the base coincide. Consider assigning either or both of these last two activities for homework. Exit Slip 2.3.2 assesses students’ understanding of the Isosceles Triangle Theorem and its converse. Journal Entry What information would convince you that a triangle is isosceles? There are several answers to this question, so see if you can find at least two of them. Look for these possible responses: (1) two sides are known to be congruent, (2) two angles are known to be congruent, (3) it has a line of symmetry. Closure Notes Display the file Unit2_Inv3_Closure.ggb on the overhead. Students should observe that the two measured angles, α and β , are unequal. Show that point D may be moved along ray ´ AC . Ask students to predict what will happen to the lengths of sides ´ AD and ´ BD of ADB as D is moved toward point C so that α = β . Move C to make the angles as nearly equal as you can. They should observe that the sides AD and BD will also be equal. Ask them to identify the theorem that this illustrates. Vocabulary Acute triangle Altitude (of triangle) Base (of isosceles triangle) Base angle (of isosceles triangle) Equilateral triangle Isosceles triangle Leg (of isosceles triangle) Median (of triangle) Obtuse triangle Right triangle Scalene triangle Vertex angle (of isosceles triangle) Unit 2 Investigation 3 Overview Connecticut Core Geometry Curriculum v 1.0
Theorems Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite these sides are congruent. Isosceles Triangle Converse: If two angles of a triangle are congruent, then the sides opposite these angles are congruent. Equilateral Triangle Theorem: If all three sides of a triangle are congruent, then all three angles are congruent. Equilateral Triangle Converse: If all three angles of a triangle are congruent, then all three sides are congruent. Resources and Materials Template for Activity 2.3.2 Compass Straightedge Scissors Graph Paper Geogebra sketch: Unit2_Inv3_Closure.ggb Activity 2.3.1 Triangles in the Coordinate Plane Activity 2.3.2 Angles in Isosceles Triangles Activity 2.3.3 Proving the Isosceles Triangle Theorem Activity 2.3.4 Proving the Isosceles Triangle Converse Activity 2.3.5 Converses of Conditional Statements (may be saved until Unit 3 Investigation 3) Activity 2.3.6 Equilateral Triangles Activity 2.3.7 Altitudes and Medians Exit Slip 2.3.1 Exit Slip 2.3.2