SSS Congruence

Show your work for each problem.

1.
Which information is enough to prove two triangles congruent by SSS?
1. a)Two pairs of equal sides
2. b)Three pairs of equal angles
3. c)Three pairs of equal sides
4. d)Equal perimeters
Answer: Three pairs of equal sides
SSS needs three pairs of congruent sides. Equal perimeters or two pairs of sides are not enough, and three equal angles only prove similarity.
2.
$\triangle ABC \cong \triangle DEF$ by SSS. $AB = 7$ , $BC = 9$ and $CA = 4$ . How long is $EF$ ?
Answer: 9
The letters in the same positions correspond: $B \leftrightarrow E$ and $C \leftrightarrow F$ . So $EF = BC = 9$ .
3.
One triangle has sides 5 cm, 6 cm and 8 cm. Another has sides 6 cm, 8 cm and 5 cm. Are they congruent?
1. a)Only if both are right triangles
2. b)No, the sides are in a different order
3. c)No, an angle is also needed
4. d)Yes, by SSS
Answer: Yes, by SSS
Both triangles have the sides 5, 6 and 8, only listed in a different order. Three pairs of equal sides, so they are congruent by SSS.
4.
Find the length of the side from $A(1, 2)$ to $B(4, 6)$ .
Answer: 5
$AB = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ .
5.
In $\triangle ABC$ : $AB = 6$ , $BC = 10$ , $CA = 8$ . In $\triangle PQR$ : $PQ = 8$ , $QR = 10$ , $RP = 6$ . Check whether the triangles are congruent by SSS.
Answer: 3
1. What is the length of the shortest side of $\triangle ABC$ ? 6
2. What is the length of the shortest side of $\triangle PQR$ ? 6
3. What is the length of the longest side in both triangles? 10
4. How many pairs of equal sides are there? 3
6.
$\triangle ABC \cong \triangle DEF$ by SSS. $DE = 12$ , $EF = 15$ and $FD = 9$ . What is the perimeter of $\triangle ABC$ ?
Answer: 36
$AB = 12$ , $BC = 15$ , $CA = 9$ , so the perimeter is $12 + 15 + 9 = 36$ .
7.
In $\triangle KLM$ : $KL = 6$ , $LM = 10$ , $MK = 7$ . In $\triangle STU$ : $ST = 10$ , $TU = 7$ , $US = 6$ . Which congruence statement is correct?
1. a) $\triangle KLM \cong \triangle STU$
2. b) $\triangle KLM \cong \triangle SUT$
3. c) $\triangle KLM \cong \triangle UST$
4. d) $\triangle KLM \cong \triangle TUS$
Answer: $\triangle KLM \cong \triangle UST$
Pairs: $KL = US = 6$ , $MK = TU = 7$ , $LM = ST = 10$ . $K$ (sides 6 and 7) matches $U$ , $L$ (sides 6 and 10) matches $S$ , $M$ (sides 7 and 10) matches $T$ . So $\triangle KLM \cong \triangle UST$ .
8.
A side of a triangle goes from $C(-2, 3)$ to $D(4, -5)$ . How long is it?
Answer: 10
$CD = \sqrt{(4 - (-2))^2 + (-5 - 3)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ .
9.
$\triangle ABC$ has $A(0, 0)$ , $B(3, 0)$ , $C(0, 4)$ . $\triangle DEF$ has $D(5, 1)$ , $E(5, 4)$ , $F(1, 1)$ . Use the distance formula to check SSS.
Answer: $\triangle ABC \cong \triangle DEF$
1. How long is $BC$ , from $(3, 0)$ to $(0, 4)$ ? 5
2. How long is $DE$ , from $(5, 1)$ to $(5, 4)$ ? 3
3. How long is $DF$ , from $(5, 1)$ to $(1, 1)$ ? 4
4. How long is $EF$ , from $(5, 4)$ to $(1, 1)$ ? 5
10.
$\triangle ABC \cong \triangle DEF$ by SSS. $AB = 3x - 4$ , $DE = x + 6$ , $BC = 2y + 1$ and $EF = 15$ . Find $x$ , $AB$ and $y$ .
Answer: $x = 5$ , $AB = 11$ , $y = 7$
1. Solve $3x - 4 = x + 6$ . What is $x$ ? 5
2. How long is $AB$ ? 11
3. Solve $2y + 1 = 15$ . What is $y$ ? 7
11.
A square and a rhombus both have four sides of length 4, but they are not congruent. Why does this not contradict SSS?
1. a)Quadrilaterals are not rigid: equal sides do not fix the angles
2. b)Their sides are not really equal
3. c)SSS only works for right triangles
4. d)A rhombus has five sides
Answer: Quadrilaterals are not rigid: equal sides do not fix the angles
SSS is a criterion for triangles only. A quadrilateral is not rigid: with the same four sides its angles can still change, so a square can be pushed into a rhombus.
12.
In the parallelogram $ABCD$ , $AB = CD = 7$ , $BC = DA = 8$ and the diagonal $AC = 13$ . $\angle ABC = 120°$ . Show that $\triangle ABC \cong \triangle CDA$ and find $\angle CDA$ .
Answer: $\angle CDA = 120°$
1. How long is the side that both triangles share? 13
2. How many pairs of equal sides do $\triangle ABC$ and $\triangle CDA$ have? 3
3. How many degrees is $\angle CDA$ ? 120
13.
$\triangle ABC \cong \triangle DEF$ by SSS, with $D(1, 2)$ and $E(7, 10)$ . How long is $AB$ ?
Answer: 10
$DE = \sqrt{(7 - 1)^2 + (10 - 2)^2} = \sqrt{36 + 64} = 10$ . Since $AB$ and $DE$ are corresponding sides, $AB = 10$ .
14.
$\triangle ABC$ has $A(1, 1)$ , $B(4, 2)$ , $C(2, 5)$ and $\triangle PQR$ has $P(6, -1)$ , $Q(5, -4)$ , $R(2, -2)$ . Compare the squared side lengths to decide whether $\triangle ABC \cong \triangle PQR$ .
Answer: $\triangle ABC \cong \triangle PQR$
1. What is $AB^2$ ? 10
2. What is $BC^2$ ? 13
3. What is $CA^2$ ? 17
4. What is $PQ^2$ ? 10
5. What is $QR^2$ ? 13
6. What is $RP^2$ ? 17
15.
Which triangle is congruent to the triangle with vertices $(0, 0)$ , $(4, 0)$ and $(0, 3)$ ?
1. a) $(0, 0)$ , $(5, 0)$ , $(0, 3)$
2. b) $(0, 0)$ , $(6, 0)$ , $(0, 8)$
3. c) $(1, 1)$ , $(5, 1)$ , $(1, 5)$
4. d) $(2, 2)$ , $(2, 6)$ , $(5, 2)$
Answer: $(2, 2)$ , $(2, 6)$ , $(5, 2)$
The given triangle has sides 4, 3 and $\sqrt{16 + 9} = 5$ . The triangle $(2, 2)$ , $(2, 6)$ , $(5, 2)$ has sides 4, 3 and $\sqrt{9 + 16} = 5$ , so it is congruent by SSS. The others have sides 6, 8, 10 (too big), 4, 4, $\sqrt{32}$ and 5, 3, $\sqrt{34}$ .
16.
$\triangle PQR \cong \triangle STU$ by SSS. $PQ = 2x + 3$ , $ST = 4x - 7$ , $QR = x + y$ , $TU = 12$ and $RP = SU = 10$ . Find the perimeter of $\triangle PQR$ .
Answer: 35
1. Solve $2x + 3 = 4x - 7$ . What is $x$ ? 5
2. How long is $PQ$ ? 13
3. Use $QR = TU$ : $5 + y = 12$ . What is $y$ ? 7
4. What is the perimeter $PQ + QR + RP$ ? 35
17.
In the kite $ABCD$ , $AB = AD$ and $CB = CD$ . $\angle BAD = 76°$ and $\angle ABC = 110°$ . Use SSS on $\triangle ABC$ and $\triangle ADC$ to find $\angle BAC$ and $\angle ADC$ .
Answer: $\angle BAC = 38°$ , $\angle ADC = 110°$
1. How many pairs of congruent sides do $\triangle ABC$ and $\triangle ADC$ have, counting the shared side? 3
2. $\angle BAC + \angle DAC$ is the whole angle $\angle BAD$ . How many degrees is that? 76
3. The two angles are equal (corresponding parts). How many degrees is $\angle BAC$ ? 38
4. How many degrees is $\angle ADC$ ? 110
18.
A square frame with sides of 5 cm has hinges at its corners. It is pushed into a rhombus with an angle of $60°$ . Explain with numbers why SSS cannot work for quadrilaterals and how a brace fixes it.
Answer: $20$ cm, $5$ cm, $2$
1. What is the perimeter of the square in cm? 20
2. What is the perimeter of the rhombus in cm? 20
3. How long is the short diagonal of the rhombus in cm? 5
4. A brace along one diagonal splits the frame into how many triangles? 2