Download Unit 1 Lines, Angles, and Triangles 1.4 Congruent Triangles

Unit 1
Lines, Angles, and Triangles
1.4 Congruent Triangles
Definitions and Notation
Congruent triangles
≅
a, b, c
a′,b′,c′
Triangles whose vertices can be made to
coincide (triangles with the same shape and
size): angles that coincide are called
corresponding angles; sides that coincide are
called corresponding sides.
Is congruent
The lengths of the sides of a triangle
opposite angles A, B, and C, respectively
A prime, b prime, c prime; the lengths of the
sides of a triangle opposite angles a′,b′, and
c′ respectively
Example
Side included by two angles of a
∆ABC ≈ ∆A′B′C′
If ∆ABC and ∆A′B′C′ are cut out, vertices A
and A′, B and B′, and C and C′ can be made
to coincide.
Corresponding sides : AB and A′B′ , BC
and B′C′ , CA and C′A′
Corresponding angles:
∠A and ∠A′, ∠B and ∠B′, ∠C and ∠C′
A side whose endpoints are vertices of two
1
Unit 1
Lines, Angles, and Triangles
1.4 Congruent Triangles
triangle
Angle included by two sides of a
triangle
angles of the triangle
An angle whose sides are two sides of the
triangle
AB is included by ∠A and ∠B
BC is included by ∠B and ∠C
CA is included by ∠A and ∠C
∠A is included by AB and CA
∠B is included by AB and BC
∠C is included by BC and CA
Properties
1. If triangles are congruent, the
corresponding sides and the
corresponding angles are equal.
If ∆ABC ≅ ∆A′B′C′
a = a′, b = b′, and c = c′, and
∠A = ∠A ′, ∠B = ∠B ′, and ∠C = ∠C ′
2. If two sides and the included angle of
one triangle are equal respectively to
two sides and the included angle of
another triangle, the triangles are
congruent. (S.A.S)
∆ABC ≅ ∆A′B′C′ if
b = b′, ∠A = ∠A ′, c = c′ or
a = a′, ∠B = ∠B ′, c = c′, or
a = a′, ∠C = ∠C ′, b = b′
3. If two angles and the included side of
one triangle are equal respectively to
two angles and the included side of
another triangle, the triangles are
congruent. (A.S.A)
∆ABC ≅ ∆A′B′C′ if
∠A = ∠A ′, c = c′ ∠B = ∠B ′, or
∠B = ∠B ′, a = a′, ∠C = ∠C ′, or
∠C = ∠C ′, b = b′, ∠A = ∠A ′
4. If the three sides of one triangle are
equal respectively to the three sides of
another triangle, the triangles are
congruent. (S.S.S.)
∆ABC ≅ ∆A′B′C′ if
a = a′, b = b′, and c = c′
5. Corresponding parts of congruent
triangles are equal. (C.P.C.T.E)
Exercises
In exercises 1-8, ∆ABC ≅
∆A′B′C′
2
Unit 1
Lines, Angles, and Triangles
1.4 Congruent Triangles
Example:
Given:
m∠A =50°, m∠ B′ =45°, a = 9 in, b = 8 in, and c′ = 10 in
Find m∠A, m∠C, m∠ C′, c′, a′, and b′
Solution:
A sketch of the triangles labeled
with the given data is useful. From
Property 1, w observe that
m∠ B = m∠ B′ = 45°. Then, since
m∠ A + m∠ B + m∠C = 180°
we have
50°+ 45°+ m∠C = 180°
m∠C = 180°- 95°
m∠C = 85°
Again, using Property 1, we observe
that
m∠C = m∠ C′ = 85°, c = c′= 10 in,
b = b = 8 in and a = a′ = 9 in
1. Given that m∠ A= 100°, m∠ B= 30°, a = 18 cm, b = 9 cm, and c′ = 15 cm.
Find all other angles and sides.
2. Given that m∠ A= 35°, m∠ B= 60°, c = 23 ft, b = 20 ft.
Find all other angles, b′ and c′.
3. Given that a = b = c = 10 mm.
Find all other angles and sides.
4. Given that a = b = 8 yd and m∠ A= 60°. Find m∠C′ and c′.
5. Given that m∠ A= 45°, m∠ B= 30°, and a = 5 m.
Find c′ and m∠B′.
6. Given that m∠ b= 45°, m∠ C= 90°, and c = 4 2 .
Find a′.
7. Given that m∠ A= 60°, m∠ C= 90°, and c = 10 ft.
Find m∠B′, b′, and a′.
8. Given that m∠ A= 30°, m∠ C= 90°, a = 8 cm.
Find b′ and c′.
In each exercise determine whether the given pairs of triangles are congruent as a
consequence of Properties 2, 3, or 4 (SAS, ASA, or SSS). When the triangles are
congruent, list all pairs of equal sides and angles other than those given. Like markings on
the parts of two triangles indicate that those parts are equal, and segments with the same
number of arrowheads pointing in the same direction indicate parallel line segments. This
symbolism will be used when convenient henceforth.
Examples:
3
Unit 1
Lines, Angles, and Triangles
1.4 Congruent Triangles
a.
Solution
The like markings indicate that ∠B = ∠F, AB = EF, and ∠A = ∠E. Therefore,
∆ABC ≅ ∆EFD by the ASA property. By property 1, the other pairs of equal
corresponding parts are ∠C = ∠D, AC = DE and BC = DF .
b.
Solution
∠BCA = ∠DCE by the property of Section 1.1. The like markings indicate that BC =
CD and AC = CE. Hence, ∆ABC ≅ ∆EDC by the SAS property. By Property 1, the
other pairs of equal corresponding parts are ∠B = ∠D, ∠A = ∠E, and AB = DE .
9.
10.
4
Unit 1
11.
Lines, Angles, and Triangles
1.4 Congruent Triangles
12.
13.
14.
15.
16.
17.
5
Unit 1
18.
Lines, Angles, and Triangles
1.4 Congruent Triangles
19.
20.
21.
22.
6