Geometric Twins · Lesson 8 of 10
Using Congruence to Prove Equal Parts
“Build clear geometry proofs from given facts, a sufficient condition, and the correct matching of parts.”
• Separate given facts from the equality that must be proved. • Use shared sides, side markings, and angle facts to establish congruence. • Prove that a kite diagonal divides two angles into equal parts. • Use parallel-line angles to prove congruence and congruence to prove parallel lines. • Combine smaller angle equalities to identify matching triangles.
From information to a conclusion
A proof explains why a conclusion follows from the information supplied. A diagram helps us locate that information, but its appearance is not a proof. First identify two triangles containing the parts you want to compare. Next collect equalities that satisfy SSS, SAS, ASA, AAS, or RHS. Finally read the required equality from the resulting vertex match.
Side markings use a convention: sides with one tick have equal lengths, sides with two ticks have equal lengths, and so on. Different tick counts do not claim an equality. A side shared by two triangles is equal to itself, whether or not a tick is drawn. A shared side often supplies the missing third fact.
| Stage | What to record |
|---|---|
| Given | The side, angle, midpoint, or parallel-line facts supplied. |
| Reason | Why each equality holds: given, shared side, angle fact, or definition. |
| Congruence | Three sufficient facts, the condition, and a correctly ordered triangle statement. |
| Conclusion | The required corresponding side or angle equality. |
A kite diagonal divides angles equally
Consider a kite-shaped figure ABCD with AB = AD and CB = CD. Draw AC. It belongs to both triangles ABC and ADC. The two given pairs together with this shared side establish SSS. Once we match B with D, the angles on either side of AC at A and at C also match.
Problem
Given AB = AD and CB = CD, show that AC divides angles BAD and BCD into equal parts.
- 1.In triangles ABC and ADC, AB = AD and BC = DC are given; AC = AC is the shared side.
- 2.Therefore ABC ≅ ADC by SSS, with A ↔ A, B ↔ D, and C ↔ C.
- 3.Corresponding angles BAC and DAC are equal. These are the two parts of angle BAD.
- 4.Corresponding angles BCA and DCA are equal. These are the two parts of angle BCD. Thus AC bisects both angles.
A line or ray that divides an angle into two equal angles.
Problem
DF = DG and FE = GE. Triangles DFE and DGE share DE. Are they congruent, and is DFE ≅ GED the correct order?
- 1.DF = DG, FE = GE, and DE = DE give all three matching side pairs.
- 2.The match is D ↔ D, F ↔ G, and E ↔ E. Hence DFE ≅ DGE by SSS.
- 3.The two triangles are congruent, but DFE ≅ GED would claim a different pairing. Correcting the order is necessary before reading corresponding parts.
Parallel lines supply equal angles
A transversal is a line that crosses two other lines. When those two lines are parallel, alternate interior angles are equal. In the crossing-diagonal figure below, AB is parallel to CD and AB = CD. The lines AC and BD are transversals, and their equal alternate angles give ASA for the top and bottom triangles.
Problem
In this figure AB ∥ CD and AB = CD. Find further equal parts.
- 1.Angle BAO = angle DCO because AC crosses the parallel lines AB and CD.
- 2.Angle ABO = angle CDO because BD crosses those same parallel lines.
- 3.AB = CD is the side included between these angle pairs, so ABO ≅ CDO by ASA.
- 4.The match is A ↔ C, B ↔ D, O ↔ O. Hence AO = CO and BO = DO. Their angles at O also agree.
The reasoning can run in the other direction. Equal segments and vertically opposite angles may establish congruence first. Corresponding angles then turn out to be equal alternate interior angles, which proves that two lines are parallel. Keep the direction of the reasoning clear: do not use the parallelism as a fact when it is the conclusion you are trying to prove.
Problem
AD and BC intersect at O. Given OA = OD and OB = OC, prove AB ∥ CD.
- 1.Angles AOB and DOC are vertically opposite and equal. Together with OA = OD and OB = OC, this gives AOB ≅ DOC by SAS.
- 2.The correspondence gives angle BAO = angle CDO.
- 3.AD crosses the lines AB and CD, and these are its alternate interior angles.
- 4.Equal alternate interior angles imply that AB and CD are parallel. Therefore AB ∥ CD.
In the ASA figure, the crossing segments are AC and BD. In the last example, they are AD and BC. Draw the stated segment names before transferring a correspondence from one problem to another.
Shared sides and combined angle facts
Sometimes a shared side lies between two angle pairs already given as equal. If angle ABC = angle DBC and angle ACB = angle DCB, triangles ABC and DBC have the same angles at B and C and share BC. They are congruent by ASA, so AB = DB, AC = DC, and angle BAC = angle BDC. The third-angle equality could also be found directly by subtracting the same two angles from 180°.
Problem
In the last figure, angle ABD = angle DCA and angle ACB = angle DBC. Identify equal sides.
- 1.Angle ABC = angle ABD + angle DBC. Angle DCB = angle DCA + angle ACB.
- 2.Each piece in the first sum equals its matching piece in the second sum, so angle ABC = angle DCB.
- 3.We also have angle ACB = angle DBC and the shared side BC = CB. Thus ABC ≅ DCB by ASA.
- 4.The order pairs A ↔ D, B ↔ C, and C ↔ B. Therefore AB = DC and AC = DB; the angles at A and D also agree.
Do not use the desired equality as a premise. Establish congruence from independent given or proved facts, then conclude that corresponding parts are equal.
Quiz
Why may AC = AC be used in the kite proof?
Which condition proves ABC ≅ ADC for AB = AD and BC = DC?
For DF = DG, FE = GE, and common DE, which order preserves the matching?
Which fact supplies angle equality when a transversal crosses parallel lines?
What follows from ABC ≅ DCB in the final example?
Practice Problems
- Draw the kite with AB = AD and CB = CD. Prove ABC ≅ ADC and state both angle-bisector equalities.
- Draw DFE and DGE sharing DE, with DF = DG and FE = GE. Give the three side reasons and the correct congruence statement.
- For AB ∥ CD and AB = CD with AC and BD crossing at O, prove ABO ≅ CDO and list all three angle pairs.
- For AD and BC crossing at O with OA = OD and OB = OC, prove AB ∥ CD without assuming it at the start.
- Given angle ABC = angle DBC and angle ACB = angle DCB, prove the triangles congruent and explain why their third angles agree.
- In the final diagram, use the two given small-angle equalities to prove ABC ≅ DCB. List every corresponding side and angle pair.
Key Takeaways
• A proof moves from given facts to a sufficient condition to corresponding-part equalities. • A shared side can complete SSS or ASA. • The kite diagonal bisects two angles because its two triangles are congruent. • Parallel lines can supply equal angles, and proved angle equalities can establish parallelism. • Add angle pieces when the angle required by a condition is not given directly.