Download 4.1 Congruent Figures (Page 198

4.1 Congruent Figures (Page 198- 203)
Congruent figures- two or more figures with the same size and shape
There are three ways to show congruency by using overlapping figures.
Congruent polygons- all their corresponding parts are congruent
Congruence statement- tells that two figures are congruent by naming their lettered
angles. For example, △ RBW @ △HJG
Theorem 4-1 (AKA the “Triangle Third Angle Congruency Theorem”)If two angles of a triangle are congruent to two angles of another triangle,
then the third angles are congruent.
This theorem can be applied in a proof like this:
Given: PQ ≅ PS, QR ≅ SR, ÐQ ≅ ÐS, ÐQPR ≅ ÐSPR
Prove: △PQR ≅ △PSR
Steps
1. PQ ≅ PS, QR ≅ SR
Reason
1. Given
Explanation
1. We only need this
much of the given
right now, we’ll use
the rest later
2. PR ≅ PR
2. Reflexive property
of Congruence
2. PR is congruent to
itself.
3. ÐQ ≅ ÐS, ÐQPR ≅
ÐSPR
3. Given
3. Here we use the
rest of the given
4. Triangle Third
Angle Congruency
Theorem
4. We already know
that the other
angles are
congruent so we
can say that those
two are congruent.
5. Definition of
Congruent
Triangles
5. Think back to the
given. Now we
have all 6 parts of
both triangles, so
we can prove them
both congruent.
4. ÐQRP ≅ ÐSRP
5. △PQR ≅ △PSR
Here are some other problems using congruent figures:
Can you conclude the figures are congruent? Explain why.
Yes. TK ≅ TK because of the “reflexive property of congruence.”
TKU ≅ TKR because of the “triangle third angle congruency
theorem.” Therefore, all sides and angles are congruent so the
triangles are congruent by the definition of congruent triangles.
Write a congruence statement for the pair of triangles.
△JYB @ △XCH
(this is a problem where it’s imperative to remember to
put the letters in the right order)
Now try on your own:
****Before you start the proof, here are some reminders:
1. When naming two figures, put letters in the same order!
2. In this section you haven’t learned AAS, SSS, and so on, so don’t use
them to solve the proof!
1. Easy
△WYS @ △MKV. List the congruent corresponding parts. Use three letters (in
the right order!) to name each angle.
2. Medium
Can you conclude that the figures are congruent? Justify your answer.
3. Hard
PROOF
Given: ÐA @ ÐE, ED @ AB , EC CD AF , FB @ CD
Prove: △AFB @ △ECD
ANSWER KEY
1. Question: △WYS @ △MKV. List the congruent corresponding parts. Use
three letters (in the right order!) to name each angle.
Answer: ÐW @ ÐY, ÐS @ ÐM, ÐK @ ÐV, WY @ MK , YS @ KV , SW @ VM
2. Question: Can you conclude that the figures are congruent? Justify your
answer.
Answer: Yes. All three sides are congruent (CA congruent to AC because
of reflexive). Angle ACD is congruent to Angle CAB because AB and DC
are parallel so they are vertical angles.
3. Question:
Given: ÐA @ ÐE, ED @ AB , EC CD AF , FB @ CD
Prove: △AFB @ △ECD
Answer:
1. ÐA @ ÐE, ED @ AB , EC CD AF ,
1. Given
FB @ CD
2. ÐC @ ÐF
3. ÐD @ ÐB
4. △AFB @ △ECD
2. Right angle congruency theorem
3. Third angle congruency theorem
4. Definition of congruent triangles