1. Geometric Twins

1. Check if the two figures are congruent.

Measure the two arm lengths and the angle between them in both figures. They are equal, so the figures are congruent (one is just turned).

2. Circle the pairs that appear congruent.

The drops (one is rotated) and the leaves (one is flipped) are congruent.

The clouds and the stars are not congruent, because one figure is smaller than the other.

3. What measurements would you take for a congruent (a) circle, (b) rectangle?

(a) Circle: its radius. Two circles are congruent if their radii are equal.

(b) Rectangle: its length and breadth. Two rectangles are congruent if their lengths are equal and their breadths are equal.

4. How would we check if two such figures are congruent?

Measure the length of the long line on each side of the joint, the length of the short arm, and the angle at the joint. If all these are equal, the figures are congruent.

Measuring the pairs in the book, both pairs are congruent.

1. ∆HEN ≅ ∆BIG. List all the other correct ways.

∆HNE ≅ ∆BGI, ∆EHN ≅ ∆IBG, ∆ENH ≅ ∆IGB, ∆NHE ≅ ∆GBI, ∆NEH ≅ ∆GIB

2. Are ∆RED and ∆JAM congruent?

RE = JA = 3.5 cm, ED = AM = 5 cm, RD = JM = 6 cm. By the SSS condition, they are congruent: ∆RED ≅ ∆JAM.

3. AB = AD and CB = CD. Find congruent triangles. Does AC divide ∠BAD and ∠BCD equally?

∆ABC and ∆ADC: AB = AD, CB = CD, and AC is common. So, ∆ABC ≅ ∆ADC (SSS).

Yes. ∠BAC = ∠DAC and ∠BCA = ∠DCA, as they are corresponding angles of congruent triangles.

4. Are ∆DFE and ∆GED congruent? (DF = DG, FE = GE)

DF = DG, FE = GE and DE is common. So, the triangles are congruent by SSS.

But the correct way to write it is ∆DFE ≅ ∆DGE (D ↔ D, F ↔ G, E ↔ E). Writing ∆DFE ≅ ∆GED is wrong, because D does not correspond to G.

1. Are the triangles congruent?

BA = ZX = 7 cm, ∠B = ∠Z = 47°, BC = ZY = 5 cm. By SAS, ∆ABC ≅ ∆XZY.

2. CD ∥ AB and AB = CD. Find the other equal parts.

∠ODC = ∠OBA and ∠OCD = ∠OAB (alternate angles). With CD = AB, by ASA, ∆DOC ≅ ∆BOA.

So, OD = OB and OC = OA. Also ∠DOC = ∠BOA (vertically opposite).

3. ∠ABC = ∠DBC and ∠ACB = ∠DCB. Show that ∠BAC = ∠BDC. Are the triangles congruent?

BC is common. By ASA, ∆ABC ≅ ∆DBC. So, ∠BAC = ∠BDC (corresponding angles).

(Also, by angle sum, ∠BAC = 180° − ∠ABC − ∠ACB = 180° − ∠DBC − ∠DCB = ∠BDC.)

4. ∠ABD = ∠DCA and ∠ACB = ∠DBC. Identify the equal parts.

∠ABC = ∠ABD + ∠DBC = ∠DCA + ∠ACB = ∠DCB. BC is common, and ∠ACB = ∠DBC.

By ASA, ∆ABC ≅ ∆DCB. So, AB = DC, AC = DB and ∠BAC = ∠CDB.

1. ∆AIR ≅ ∆FLY. Corresponding parts

Vertices: A and F, I and L, R and Y.

Sides: AI and FL, IR and LY, AR and FY.

Angles: ∠A and ∠F, ∠I and ∠L, ∠R and ∠Y.

2. Identify the congruent pairs.

(a) AB = DE, BC = EF, CA = FD → SSS: ∆ABC ≅ ∆DEF.

(b) AB = EF, ∠A = ∠E, AC = ED → SAS: ∆ABC ≅ ∆EFD.

(c) ∠B = ∠D = 90°, AB = DF, hypotenuse AC = FE → RHS: ∆ABC ≅ ∆FDE.

(d) ∠A = ∠D, ∠B = ∠E, AC = DF → AAS: ∆ABC ≅ ∆DEF.

(e) AB = DF, ∠B = ∠F, AC = DE → this is SSA. The triangles need not be congruent.

3. OB = OC and OA = OD. Show that AB ∥ CD.

∠AOB = ∠DOC (vertically opposite). So, ∆AOB ≅ ∆DOC (SAS).

Then ∠OAB = ∠ODC. These are alternate angles with the transversal AD. So, AB ∥ CD.

4. ABCD is a square. Show ∆ABC ≅ ∆ADC. Is ∆ABC ≅ ∆CDA?

AB = AD, BC = DC and AC is common. So, ∆ABC ≅ ∆ADC (SSS).

Also AB = CD, BC = DA and CA = AC. So, yes, ∆ABC ≅ ∆CDA too.

Another example: an isosceles triangle ABC with AB = AC is congruent to itself as ∆ABC ≅ ∆ACB.

Six ways: two equilateral triangles with the same side. Any vertex can match any vertex, giving 6 correspondences.

5. A is the centre of the circle and ∠A = 120°. Find ∠B and ∠C.

AB = AC (radii). So, ∠B = ∠C = (180° − 120°) ÷ 2 = 30° each.

6. Find the missing angles.

Name the inner points P (below R), Q (where the 44°, 46° and 90° are), M (lower left) and N (lower right). Some of the missing angles are: