Another way to write that triangle ABC is congruent to triangle DEF is like this:
Notice that the triangles are exactly the same size and shape -
First, look at the lengths:
AB = DE
BC = EF
AC = DF
All of the corresponding sides are equal.
Now, look at the angles:
< ABC = < DEF
< BCA = < EFD
< CAB = < FDE
All of the corresponding angles are equal.
Note: Before finding the measurements for the missing sides or angles of congruent triangles, it is first necessary to state which angles/sides are congruent, and then we must state which Condition for Congruence we are using:
Conditions for Congruence
There are three different ways that we can tell whether or not triangles are congruent. They involve looking at the side lengths, as well as the angles.
1. Side Side Side (SSS)
All three of the corresponding sides are equal.
For example:
AB = DE
BC = EF
AC = DF
2. Side Angle Side (SAS)
Two pairs of corresponding sides and the contained angle (the angle between the sides) are equal.
For example:
AB = DE
BC = EF
< ABC = < DEF
3. Angle Side Angle
two pairs of corresponding angles and the contained side (the side between the angle) are equal.
For example:
AB = DE
< ABC = < DEF
< BAC = < EDF
Because angle ACB and angle EBD are vertically opposite, they are the same angle (in this case, 90°).
And, the question is telling us that side AC is equal to side CE.
Finally, because AB and DE are parallel, we can use the Z-pattern to find that angle BAC = DEC.
Therefore:
< ACB = < EBD
AC = CE
< BAC = < DEC
By Angle-Side-Angle, triangles CAB and CED are congruent. Note: The way in which the sides are named is important!
Because < BAC = 40° < DEC, < DEC is also equal to 40°.
We know that the angles inside a triangle add up to 180°.
180° - 90° - 40° = < CDE
< CDE = 50°.
We also know that BC = CD, and that BC + CD = 8 cm.
Therefore, BC and CD each equal 4 cm.
Thus, BC = 4 cm, and < CDE = 50°.
Practice Questions
Page 462 #5, 6be, 7ab,
