Prove congruence and use it to prove a further result

AB = AD

\(\displaystyle 5.8\ \mathrm{cm}\)

BC = DC

\(\displaystyle 6.3\ \mathrm{cm}\)

The two pairs of outer sides are equal. Which additional fact completes an SSS proof for triangles ABC and ADC?

Change both sliders. Compare the two outer side pairs and the shared dashed diagonal. The diagram rescales to stay readable; length readouts are rounded to two decimal places

Which three pairs stayed equal for every position of the slider, giving SSS?

Proving congruence

A congruence proof lists three matched equal parts with a reason for each, names the test, and only then uses corresponding parts for a further result

1. 1State the evidence Give a reason for each equality: a supplied fact, a common side, or an established angle relationship
2. 2Match corresponding vertices Use equal sides and angles to find the matching vertices. Their left-to-right positions in the drawing do not decide the order
3. 3Name a valid test Use SSS, SAS with the included angle, AAS/ASA, or RHS. Choose the test supported directly by the proof conditions
4. 4Draw the further conclusion After proving congruence, corresponding sides and angles are equal. Use those equalities to prove the requested result
5. 5Avoid circular reasoning Do not assume that corresponding parts are equal because the triangles are congruent while you are still trying to prove that congruence
6. 6Distinguish a rule from its converse Parallel lines give equal alternate angles. Conversely, equal alternate angles on a transversal prove that the two lines are parallel

PR and QS cross at M. Why is angle PMQ equal to angle RMS?

Parallel sides

\(\displaystyle PQ\parallel SR\)

PQ is parallel to SR, with PR as a transversal. Why is angle QPM equal to angle SRM?

M is the midpoint of PR. Which line gives the equality with the correct reason?

First angle pair

\(\displaystyle \angle QPM=\angle SRM\)

Crossing angle pair

\(\displaystyle \angle PMQ=\angle RMS\)

Midpoint fact

\(\displaystyle PM=MR\)

Using the two angle pairs and PM = MR, which correspondence completes the ASA proof?

A student tries to prove PQM congruent to RSM by saying QM = MS because corresponding sides of congruent triangles are equal. What is wrong?

PQ

\(\displaystyle 7.2\ \mathrm{cm}\)

QM

\(\displaystyle 4.9\ \mathrm{cm}\)

The angle and midpoint facts prove triangle PQM congruent to triangle RSM. Find SR.

Match the vertices: P matches R and Q matches S. Which side of the second triangle matches PQ?

QM

\(\displaystyle 4.9\text{ cm}\)

MS corresponds to

\(\displaystyle MQ\)

Given triangle PQM is congruent to triangle RSM, find the whole diagonal QS

MS is a side of the second triangle; which side of the first does it match?

Common hypotenuse AC

\(\displaystyle 13\ \mathrm{cm}\)

AB = AD

\(\displaystyle 5\ \mathrm{cm}\)

Angles ABC and ADC are right angles. The triangles are congruent by RHS. Find DC

Use Pythagoras: the missing leg squared is 13² − 5²

A completed proof gives triangle ABC congruent to triangle CDA. The triangles lie on opposite sides of AC. Since angle BAC = angle DCA, what follows?

Equal sides

\(\displaystyle AB=AC\)

Vertex angle BAC

\(\displaystyle 80^\circ\)

In a triangle used for a SAS proof, AB = AC and angle BAC = 80°. Find angle ABC

The two base angles are equal and together total 180° − 80°

A triangle used in an SSS proof has sides 6 cm, 8 cm and 10 cm. What is the angle opposite the 10 cm side?

Two triangles share side AC. A proof uses AC = AC with the reason “vertically opposite angles”. How should it be corrected?