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Lesson 10 of 10

Geometric Twins · Lesson 10 of 10

Chapter Summary and Practice

“Connect all five congruence conditions, revisit angle properties, and solve mixed proofs and visual challenges.”

Learning Objectives

• Choose and justify an appropriate congruence condition. • Write correct vertex, side, and angle correspondences. • Recognise insufficient AAA or generic SSA information. • Solve mixed congruence proofs and equal-side angle problems. • Use congruence and angle facts in a larger diagram and a grid challenge.

Choose enough information to fix a triangle

Congruence means an exact match in shape and size, allowing a slide, a turn, or a flip. A circle is fixed by its radius, and a rectangle by its length and breadth. A bent two-arm symbol needs both arm lengths and their opening angle. For a triangle, several different combinations of measurements can fix an exact copy. The important question is whether the information removes all freedom to change its shape or size.

ConditionSufficient corresponding informationWhat to check
SSSThree sidesPair all three lengths with the correct endpoints.
SASTwo sides and their included angleThe angle is at the common endpoint of those sides.
ASATwo angles and their included sideThe side joins the two given angle vertices.
AASTwo angles and a non-included sideUse the third-angle sum to obtain ASA.
RHSA right angle, hypotenuse, and one legBoth triangles are right triangles; match hypotenuse to hypotenuse.
AAA: insufficientThree anglesAn enlargement has the same angles but a different size.
Generic SSA: insufficientTwo sides and a non-included angleThere can be two non-congruent constructions.

The conditions are supported by construction: SSS uses intersections of circles, SAS fixes two arms, ASA locates the meeting of two rays, and RHS combines a perpendicular with a circle. AAS follows by finding the missing angle. These explanations help you remember why a condition works instead of memorising its initials alone.

Sort a mixed set of measurements

Problem
For triangles ABC and a second triangle with vertices D, E, F, identify the condition and order in each case.

  1. 1.AB = DE, BC = EF, CA = DF: A ↔ D, B ↔ E, C ↔ F. Therefore ABC ≅ DEF by SSS.
  2. 2.AB = EF, AC = ED, angle A = angle E: the angle is between the two given sides. Therefore ABC ≅ EFD by SAS.
  3. 3.AB = DF, AC = FE, angle B = angle D = 90°: AC and FE are hypotenuses. Therefore ABC ≅ FDE by RHS.
  4. 4.Angle A = angle D, angle B = angle E, AC = DF: the given side is non-included between the given angles. Therefore ABC ≅ DEF by AAS.
  5. 5.AB = DF, AC = DE, angle B = angle F: these are two sides and a non-included angle. Generic SSA does not establish congruence.

Keep the correspondence throughout a proof

A statement such as AIR ≅ FLY pairs A with F, I with L, and R with Y. It gives AI = FL, IR = LY, AR = FY, and the three matching angle equalities. Reordering the first triangle requires the same change in the second. After proving congruence, use this pairing to find the required part rather than relying on how the picture happens to face.

Symmetry can allow more than one valid correspondence. A general triangle with three unequal sides usually has only one pairing to an exact copy. An isosceles triangle can allow its two base vertices to swap. An equilateral triangle permits any of the six orders of its three vertices, because all its sides and angles agree.

Two valid correspondences in a square

Problem
ABCD is a square named around its boundary. Show both ABC ≅ ADC and ABC ≅ CDA.

  1. 1.For ABC and ADC, AB = AD and BC = DC because all square sides are equal; AC is shared. Thus ABC ≅ ADC by SSS.
  2. 2.For ABC and CDA, AB = CD, BC = DA, and AC = CA. Thus ABC ≅ CDA by SSS as well.
  3. 3.The first statement pairs A ↔ A, B ↔ D, C ↔ C. The second pairs A ↔ C, B ↔ D, C ↔ A. Both preserve all side lengths.
  4. 4.The triangles are isosceles right triangles. Their equal-side symmetry allows two pairings; this does not make arbitrary pairings valid for every triangle.

Equal sides, equal angles, and proof habits

In an isosceles triangle the angles opposite the equal sides are equal. The altitude from the apex gives two right triangles congruent by RHS, which explains the property. In an equilateral triangle all three angles are 60°. Equal radii are another way to obtain equal sides. When numerical angles are needed, combine these properties with the 180° triangle angle sum.

Combine radii with the angle sum

Problem
A circle has centre O. P and Q lie on the circle, and angle POQ = 110°. Find both base angles.

  1. 1.OP = OQ because they are radii of the same circle, so angles OPQ and OQP are equal.
  2. 2.Their sum is 180° − 110° = 70°.
  3. 3.Each is 70° ÷ 2 = 35°. Check that 110° + 35° + 35° = 180°.

Before completing a proof, check what is given and what must be proved. Shared sides, midpoints, vertically opposite angles, and alternate angles can supply the missing equalities. If parallelism is the goal, prove suitable alternate angles equal before concluding the lines are parallel. If congruence is the goal, do not assume equal corresponding parts before establishing a sufficient condition.

Common mistake

An attractive or symmetric-looking sketch is not a measurement. AAA leaves size free; generic SSA can leave two shapes; and an incorrectly ordered congruence statement can attach a correct equality to the wrong part.

A connected angle challenge

The diagram below brings several chapter ideas together. Matching tick counts indicate equal lengths, and the printed angle values are given information. It is schematic, so use the marked facts rather than a protractor. Four additional letters M, N, O, and P name the interior junctions so you can describe your reasoning precisely. Solve one small triangle at a time, recording each new angle before moving to a neighbouring triangle.

ABCDURVMNOPKLF56°34°34°34°68°44°46°56°30°44°98°
Angle network with equal-side markings— Use equal-side triangles, the triangle angle sum, and straight-line angles. Do not measure this schematic drawing.
Read the network accurately

Straight boundary lines contain C–R–V–D, C–U–A, A–K–L–B, and D–F–B. A–N–O–D and K–P–F are straight lines. The given angles are UAN = 56°, NAK = 34°, MUN = 34°, MRO = 34°, RVO = 68°, ROM = 44°, MON = 46°, NOP = 90°, NPO = 56°, NPK = 30°, ANK = 44°, and DFO = 98°. The corners at C and B are right angles, and PL is perpendicular to AB. Equal-side groups are shown by their tick counts.

Practice Problems

Angle Network Challenge
  1. Use CR = CU and the right angle at C to find angles CRU and CUR.
  2. Use UM = UN and angle MUN = 34° to find angles UMN and UNM.
  3. Use RV = VO and angle RVO = 68° to find angles VRO and ROV.
  4. Find angle RMO from the two given angles in triangle RMO.
  5. Use AU = UN and angle UAN = 56° to find angles ANU and AUN.
  6. Find angle AKN from triangle ANK.
  7. Use the three double-ticked sides to find all angles of triangle PFB. Then find angle PBL using the right angle at B.
  8. Use triangle PBL and PL perpendicular to AB to find angle LPB. Compare KLP and BLP by SAS to find angles KPL and PKL.
  9. Find angle ONP in triangle NOP.
  10. Use the straight line DFB, angle DFO, and triangle PFB to find angle OFP. Use the straight line KPF and the angles at P to find angle OPF, then angle POF.
  11. Continue finding other unlabelled angles. For each answer, write the triangle, equality, or straight-line fact that justifies it.

CRU = CUR = 45° because CR = CU and angle C = 90°. UMN = UNM = (180° − 34°) ÷ 2 = 73°. VRO = ROV = (180° − 68°) ÷ 2 = 56°.

Split a grid into congruent regions

Congruence also applies to figures that are not triangles. The following five-by-five grid has its centre square removed from the white region. Split the 24 white squares into six connected congruent regions, using whole squares and drawing boundaries on grid lines. Rotations and flips are allowed when comparing the six regions. Each region must contain four squares, but equal area alone is not enough: their shapes must fit too.

Centre square excluded
Six congruent regions challenge— Divide only the white squares into six congruent connected regions. Each region must have four squares.
A useful way to investigate

Try drawing a four-square shape on spare grid paper and rotate or flip it as you place copies. Keep all white squares covered exactly once, with no overlap and no copy passing through the centre square. You may need to change the first shape you try. Verify a proposed solution by tracing one region onto each of the other five.

Number rows from top to bottom and columns from left to right, each from 1 to 5. A pair (row, column) identifies one square. Region 1: (4,1), (5,1), (5,2), (5,3). Region 2: (3,5), (4,5), (5,4), (5,5). Region 3: (1,3), (1,4), (1,5), (2,3). Region 4: (2,4), (2,5), (3,4), (4,4). Region 5: (2,2), (3,2), (4,2), (4,3). Region 6: (1,1), (1,2), (2,1), (3,1). Each region is the same four-square L shape after a turn or flip. Together they cover all 24 white squares once and exclude (3,3).

Quiz

Quick check

Which data establish SAS?

Quick check

Which statement follows from AIR ≅ FLY?

Quick check

Which information leaves the triangle’s size free?

Quick check

An isosceles triangle has apex angle 42°. What is each base angle?

Quick check

Two equilateral triangles have the same side length. How many vertex pairings are valid?

Quick check

How many white unit squares belong to each of six congruent grid regions?

Quick check

Both triangles are right triangles, with equal hypotenuses and one equal leg pair. Which condition applies?

Mixed Chapter Practice
  1. Explain congruence by describing how a tracing can be slid, turned, or flipped to match another figure.
  2. What measurements fix a circle, a rectangle, and a bent two-arm symbol? Explain why arm lengths alone are insufficient.
  3. Construct a 4 cm, 6 cm, 8 cm triangle with arcs. Explain why the two possible circle-intersection choices are congruent.
  4. Give an enlargement example showing why matching all three angles does not establish congruence.
  5. Construct AB = 6 cm, AC = 5 cm, angle A = 30°. Name the condition that fixes the triangle.
  6. Construct the SSA counterexample PQ = 6 cm, QR = 4 cm, angle P = 30°, and explain the two possible third vertices.
  7. Construct BC = 5 cm, angle B = 50°, angle C = 30°. Calculate angle A and explain ASA.
  8. Angle A = angle X = 35°, angle C = angle Z = 75°, and BC = YZ = 4 cm. Derive the third-angle equality and prove ABC ≅ XYZ.
  9. Construct a right triangle with one leg 4 cm and hypotenuse 5 cm. Identify the right vertex and explain RHS.
  10. If AIR ≅ FLY, list all corresponding parts. Write five other equivalent orders with triangle AIR on the left.
  11. Draw a rectangle ABCD with diagonal BD. Prove ABD ≅ CDB, and explain why matching B to B need not work.
  12. Draw a kite with AB = AD and CB = CD. Prove that AC bisects angles BAD and BCD.
  13. Let AD and BC intersect at O, where OA = OD and OB = OC. Prove AB = DC and then AB ∥ CD.
  14. AB = AC and angle A = 118°. Find the two base angles. Then find the apex angle of an isosceles triangle whose base angles are 74°.
  15. A is a circle’s centre, B and C lie on it, and angle BAC = 120°. Find the other two triangle angles with reasons.
  16. For a square ABCD, justify both ABC ≅ ADC and ABC ≅ CDA. Give two equal-size equilateral triangles and write their six valid correspondences.
  17. Sketch a repeating triangular design inspired by a dome, pyramid, rangoli, or bridge. Identify enough measurements to recreate one panel and prove two panels congruent.
  18. Complete the angle network and six-region grid challenge above. Check every result using reasons rather than appearance.

Key Takeaways

Key Takeaways

• Congruence fixes both shape and size, allowing slides, turns, and flips. • SSS, SAS, ASA, AAS, and RHS are sufficient when their exact requirements are met. • AAA and generic SSA alone do not guarantee congruence. • A triangle statement must preserve the correct vertex correspondence. • Congruence proofs justify equal sides, equal angles, angle bisectors, and parallel-line conclusions. • Equal-side angle properties and the 180° angle sum support numerical and visual reasoning.

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Angles of Isosceles and Equilateral Triangles

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