c. Place the compass on pointBand draw an arc across the interior of the angle.
d. Without changing the radius of the compass, place the compass on pointCand draw an arc across the interior
of the angle.
e. Label the intersection of the two arc asD. Draw−→
BD.
f.−→
BDis the bisector of^6 ABC.An example is show below.
Isosceles and Equilateral Triangles
Pacing:This lesson should take one class period
Goal:There is a natural progression from triangle congruencies to isosceles and equilateral triangles. This lesson
illustrates the special properties that arise from these two types of polygons.
It is helpful for students to reproduce the isosceles triangle drawing. They will benefit from using this diagram when
completing the exercises.
Another useful theorem of isosceles triangles has an especially long name. TheIsosceles Triangle Coincidence
Theoremstates, “If a triangle is isosceles, then the bisector of the vertex angle, the perpendicular bisector to the
base, and the median to the base are the same line.” Therefore, the perpendicular bisector to an isosceles triangle’s
base is the same line generated by the angle bisector of the vertex.
Because of this theorem, step 5 of the proof in example 1 can be alternatively justified using the HL Congruence
Theorem.ADcreates two right angles,^6 ADCand^6 ADB.
An equilateral triangle is a special type of isosceles triangle. Some definitions of isosceles state, “An isosceles
triangle hasat least twosides of equal length.” While this book does not use this exact definition, it is implied by
using the Isosceles Triangle Base Angle Theorem with equilateral triangles.
Extension!Since the Isosceles Triangle Base Angle Theorem and its converse are true, have your students create a
biconditional of this theorem.The base angles of a triangle are congruent if and only if the triangle is isosceles.
Extension!Students may fall into the trap of assuming figures must be both equiangular and equilateral. To show
a counterexample to the belief, have students create equiangular polygons that are not equilateral. For example,
students can draw a pentagon having five interior angle measurements of 108 degrees with varying side lengths.
Congruence Transformations
Pacing:This lesson should take one class period
Goal:This lesson introduces students to isometries, which can also be found in Chapter 12. The main focus of this
lesson is on congruent triangle transformations.
Name That Transformation!Create slides or cards with images of transformations (preimage and image). Flash one
at a time to the class. Have the students answer on a personal whiteboard or other monitoring system. Offer one
1.4. Congruent Triangles