SCHAUM’S outline Geometry

CHAPTER 3
Congruent Triangles

3.1 Congruent Triangles

Congruent figures are figures that have the same size and the same shape; they are the exact duplicates of each other. Such figures can be moved on top of one another so that their corresponding parts line up exactly. For example, two circles having the same radius are congruent circles.

Congruent triangles are triangles that have the same size and the same shape.

If two triangles are congruent, their corresponding sides and angles must be congruent. Thus, congruent triangles ABC and A′B′C′ in Fig. 3-1 have congruent corresponding sides (, and ) and congruent corresponding angles (∠A ≐ ∠A′, ∠B ≐ ∠B’, and ∠ C ≐ ∠ C’).

Fig. 3-1

(Read ΔABC ≐ ΔA′B′C′ as “Triangle ABC is congruent to triangle A-prime, B-prime, C-prime.”)

Note in the congruent triangles how corresponding equal parts may be located. Corresponding sides lie opposite congruent angles, and corresponding angles lie opposite congruent sides.

3.1A Basic Principles of Congruent Triangles

PRINCIPLE 1: If two triangles are congruent, then their corresponding parts are congruent. (Corresponding parts of congruent triangles are congruent.)

Thus if ΔABC ≐ ∠A′B′C′ in Fig. 3-2, then ∠A ≐ ∠A′, ∠B ≐ ∠B′, ∠ C ≐ ∠ C′, a = A′, b = b’, and c = C′.

Fig. 3-2

Methods of Proving that Triangles are Congruent

PRINCIPLE 2: (Side-Angle-Side, SAS) If two sides and the included angle of one triangle are congruent to the corresponding parts of another, then the triangles are congruent.

Thus if b = b’, c = c’, and ∠A = ∠A’ in Fig. 3-3, then ΔABC = ΔA′B′C′.

Fig. 3-3

PRINCIPLE 3: (Angle-Side-Angle, ASA) If two angles and the included side of one triangle are congruent to the corresponding parts of another, then the triangles are congruent.

Thus if ∠A ≐ ∠A′, ∠ C ≐ ∠ C’, and b = b’ in Fig. 3-4, then ΔABC ≐ Δ A′B′C′.

Fig. 3-4

PRINCIPLE 4: (Side-Side-Side, SSS) If three sides of one triangle are congruent to three sides of another, then the triangles are congruent.

Thus if a = A′, b = b’, and c = c’ in Fig. 3-5, then ΔABC ≐ ∠A′B′C.

Fig. 3-5

SOLVED PROBLEMS

3.1 Selecting congruent triangles

From each set of three triangles in Fig. 3-6 select the congruent triangles and state the congruency principle that is involved.

Solutions

(a) ΔI ≐ ΔII, by SAS. In ΔIII, the right angle is not between 3 and 4.

(b) ΔII ≐ ΔIII, by ASA. In ΔI, side 10 is not between 70° and 30°.

Fig. 3-6

3.2 Determining the reason for congruency of triangles

In each part of Fig. 3-7, ΔI can be proved congruent of ΔII. Make a diagram showing the equal parts of both triangles and state the congruency principle that is involved.

Fig. 3-7

Solutions

(a) AC is a common side of both [Fig. 3-8(a)]. ΔI ≐ ΔII by ASA.

(b) ∠1 and ∠2 are vertical angles [Fig. 3-8(b)]. ΔI ≐ ΔII by SAS.

Fig. 3-8

3.3 Finding parts needed to prove triangles congruent

State the additional parts needed to prove ΔI ≐ ΔII in the given figure by the given congruency principle.

Fig. 3-9

(a) In Fig. 3-9(a) by SSS.

(b) In Fig. 3-9(a) by SAS.

(d) In Fig. 3-9(c) by ASA.

(e) In Fig. 3-9(c) by SAS.

Solutions

(a) If ≐ , then ΔI ≐ ΔII by SSS.

(b) If ∠1 ≐ ∠4, then ΔI ≐ ΔII by SAS.

(d) If ∠2 ≐ ∠3, then ΔI ≐ ΔII by ASA.

(e) If ≐ , then ΔI ≐ ΔII by SAS.

3.4 Selecting corresponding parts of congruent triangles

In each part of Fig. 3-10, the equal parts needed to prove ΔI = ΔII are marked. List the remaining parts that are congruent.

Fig. 3-10

Solutions

Congruent corresponding sides lie opposite congruent angles. Congruent corresponding angles lie opposite congruent sides.

(a) Opposite 45°, ≐ , Opposite 80°, ≐ . Opposite the side of length 12; ∠C ≐ ∠ D.

(b) Opposite and ∠3 ≐ ∠2. Opposite and , ∠1 ≐ ∠4. Opposite common side , ∠A ≐ ∠C.

3.5 Applying algebra to congruent triangles

In each part of Fig. 3-11, find x and y.

Fig. 3-11

Solutions

(a) Since ΔI ≐ ΔII, by SSS, corresponding angles are congruent. Hence, 2x = 24 or x = 12, and 3y = 60 or y = 20.

(b) Since ΔI ≐ ΔII, by SSS, corresponding angles are congruent. Hence, x + 20 = 26 or x = 6, and y –5 = 42 or y = 47.

(c) Since ΔI ≐ ΔII, by ASA, corresponding are congruent. Then 2x = 3y + 8 and x = 2y. Substituting 2y for x in the first of these equations, we obtain 2(2y) = 3y + 8 or y = 8. Then x = 2y = 16.

3.6 Proving a congruency problem

PROOF:

3.7 Proving a congruency problem stated in words

Prove that if the opposite sides of a quadrilateral are equal and a diagonal is drawn, equal angles are formed between the diagonal and the sides.

Solution

If the opposite sides of a quadrilateral are congruent and a diagonal is drawn, congruent angles are formed between the diagonal and the sides.

PROOF:

3.2 Isosceles and Equilateral Triangles

3.2A Principles of Isosceles and Equilateral Triangles

PRINCIPLE 1: If two sides of a triangle are congruent, the angles opposite these sides are congruent. (Base angles of an isosceles triangle are congruent.)

Thus in ΔABC in Fig. 3-12, if ≐ , then ∠A ≐ ∠ C.

A proof of Principle 1 is given in Chapter 16.

Fig. 3-12

PRINCIPLE 2: If two angles of a triangle are congruent, the sides opposite these angles are congruent.

Thus in ΔABC in Fig. 3-13, if ∠A ≐ ∠C, then ≐ .

Fig. 3-13

Principle 2 is the converse of Principle 1. A proof of Principle 2 is given in Chapter 16.

PRINCIPLE 3: An equilateral triangle is equiangular.

Thus in ΔABC in Fig. 3-14, if ≐ ≐ , then ∠A ≐ ∠B ≐ ∠ C.

Fig. 3-14

Principle 3 is a corollary of Principle 1. A corollary of a theorem is another theorem whose statement and proof follow readily from the theorem.

PRINCIPLE 4: An equiangular triangle is equilateral.

Thus in ΔABC in Fig. 3-15, if ∠A ≐ ∠B ≐ ∠ C, then ≐ ≐ .

Principle 4 is the converse of Principle 3 and a corollary of Principle 2.

Fig. 3-15

SOLVED PROBLEMS

3.8 Applying principles 1 and 3

In each part of Fig. 3-16, name the congruent angles that are opposite congruent sides of a triangle.

Fig. 3-16

Solutions

(a) Since ≐ , ∠A ≐ ∠B.

(b) Since ≐ , ∠1 ≐ ∠2. Since ≐ , ∠3 ≐ ∠4.

(d) Since ≐ ≐ , ∠A ≐ ∠ACB ≐ ∠ABC. Since ≐ ≐ , ∠A ≐ ∠ D ≐ ∠ E.

3.9 Applying principles 2 and 4

In each part of Fig. 3-17, name the congruent sides that are opposite congruent angles of a triangle.

Fig. 3-17

Solutions

(a) Since m∠A = 55°, ∠A ≐ ∠ D. Hence, ≐ .

(b) Since ∠A ≐ ∠1, ≐ . Since ∠2 ≐ ∠ C, ≐ .

(d) Since ∠A Δ ∠1 ≐ ∠4, ≐ ≐ . Since ∠2 ≐ ∠ C, ≐ .

3.10 Applying isosceles triangle principles

In each of Fig. 3-18(a) and (b), ΔI can be proved congruent to ΔII. Make a diagram showing the congruent parts of both triangles and state the congruency principle involved.

Fig. 3-18

Solutions

(a) Since ≐ , ∠A ≐ ∠ C. ΔI ≐ ΔII by SAS [see Fig. 3-19(a)].

(b) Since ≐ , ∠B ≐ ∠ C. ΔI ≐ ΔII by ASA [see Fig. 3-19(b)].

Fig. 3-19

3.11 Proving an isosceles triangle problem

PROOF:

3.12 Proving an isosceles triangle problem stated in words

Prove that the bisector of the vertex angle of an isosceles triangle is a median to the base.

Solution

The bisector of the vertex angle of an isosceles triangle is a median to the base.

PROOF:

SUPPLEMENTARY PROBLEMS

3.1. Select the congruent triangles in (a) Fig. 3-20, (b) Fig. 3-21, and (c) Fig. 3-22, and state the congruency principle in each case.

(3.1)

Fig. 3-20

Fig. 3-21

Fig. 3-22

3.2. In each figure below, ΔI can be proved congruent to ΔII. State the congruency principle involved.

(3.2)

3.3. State the additional parts needed to prove ΔI ≐ ΔII in the given figure by the given congruency principle.

(3.3)

Fig. 3-23

(a) In Fig. 3-23(a) by SSS.

(b) In Fig. 3-23(a) by SAS.

(d) In Fig. 3-23(b) by SAS.

(e) In Fig. 3-23(c) by SSS.

(f) In Fig. 3-23(c) by SAS.

3.4. In each part of Fig. 3-24, the congruent parts needed to prove ΔI ≐ ΔII are marked. Name the remaining parts that are congruent.

(3.4)

Fig. 3-24

3.5. In each part of Fig. 3-25, find x and y.

(3.5)

Fig. 3-25

3.6. Prove each of the following.

(3.6)

Fig. 3-26

Fig. 3-27

Fig. 3-28

Fig. 3-29

3.7. Prove each of the following:

(3.7)

(a) If a line bisects an angle of a triangle and is perpendicular to the opposite side, then it bisects that side.

(b) If the diagonals of a quadrilateral bisect each other, then its opposite sides are congruent.

(c) If the base and a leg of one isosceles triangle are congruent to the base and a leg of another isosceles triangle, then their vertex angles are congruent.

(d) Lines drawn from a point on the perpendicular bisector of a given line to the ends of the given line are congruent.

(e) If the legs of one right triangle are congruent respectively to the legs of another, their hypotenuses are congruent.

3.8. In each part of Fig. 3-30, name the congruent angles that are opposite sides of a triangle.

(3.8)

Fig. 3-30

3.9. In each part of Fig. 3-31, name the congruent sides that are opposite congruent angles of a triangle.

(3.9)

Fig. 3-31

3.10. In each part of Fig. 3-32, two triangles are to be proved congruent. Make a diagram showing the congruent parts of both triangles and state the reason for congruency.

(3.10)

Fig. 3-32

3.11. In each part of Fig. 3-33, ΔI, ΔII, and ΔIII can be proved congruent. Make a diagram showing the congruent parts and state the reason for congruency.

(3.10)