of communication students have an easier time understanding why they must put the corresponding vertices in the
same order when writing the congruence statement.
Third Angle Theorem by Proof –In the text an example is given to demonstrate the Third Angle Theorem, this
is inductive reasoning. A deeper understanding of the theorem, and different types of reasoning, can be gained by
using deductive reasoning to write a proof. It will also reinforce the idea that theorems must be proved, and shows
how inductive and deductive reasoning work together.
Key Exercise: Prove the Third Angle Theorem.
Answer: Refer to the figures on the top of page 213, where the example of the Third Angle Theorem is given.
TABLE2.7:
Statement Reason(^6) W∼= (^6) C Given
(^6) V∼= (^6) A Given
m^6 V+m^6 W+m^6 X= 180 Triangle Sum Theorem
m^6 C+m^6 A+m^6 T= 180 Triangle Sum Theorem
m^6 V+m^6 W+m^6 X=m^6 C+m^6 A+m^6 T Substitution Property of Equality
m^6 C+m^6 A+m^6 X=m^6 C+m^6 A+m^6 T Subtraction Property of Equality
m^6 X=m^6 T Substitution Property of Equality
Triangle Congruence Using SSS
One Triangle or Two –In previous chapters, students learned to classify a single triangle by its sides. Now students
are comparing two triangles by looking for corresponding pairs of congruent sides. Evaluating the same triangle in
both of these ways helps the students remember the difference, and is a good way to review previous material. For
instance, students could be asked to draw a pair of isosceles triangles that are not congruent, and a pair of scalene
triangles that can be shown to be congruent with the SSS postulate.
Correct Congruence Statements –Determining which vertices of congruent triangles correspond is more difficult
when no congruent angles are marked. Once the students have determined that the triangles are in fact congruent
using the SSS Congruence postulate, it is advisable for them to mark congruent angles before writing the congruence
statement. Corresponding congruent angles are found by matching up side markings. The angle made by the sides
marked with one and two tick marks corresponds to the angle made by the corresponding sides in the other triangle,
and so on.
Translation Rotation –Translating a triangle on a coordinate plane in order to see if it fits exactly over another
triangle is a good way to demonstrate that two triangles are congruent. The notation used to describe these transla-
tions can sometimes be confusing. The text writes out the movement in words “Dis 7 units to the right and 8 units
belowA”. If students use other materials for reference, they may see this same translation as( 7 ,− 8 ). This could be
confused with the point located at( 7 ,− 8 ). It may be helpful to alert students to this difference.
Additional Exercises:
- Use the congruence statement and given information to find the indicated measurements.
4 ABC∼= 4 ZY X
m^6 A= 52 ◦
m^6 Y= 85 ◦
AC=12 cm2.4. Congruent Triangles