SAS Congruence

Show your work for each problem.

1.
In $\triangle ABC$ , which angle is included between the sides $AB$ and $BC$ ?
1. a)None of them
2. b) $\angle A$
3. c) $\angle C$
4. d) $\angle B$
Answer: $\angle B$
$AB$ and $BC$ meet at $B$ , so the included angle is $\angle B$ .
2.
Which information proves $\triangle ABC \cong \triangle DEF$ by SAS?
1. a) $AB = DE$ , $BC = EF$ , $CA = FD$
2. b) $AB = DE$ , $\angle B = \angle E$ , $BC = EF$
3. c) $\angle A = \angle D$ , $\angle B = \angle E$ , $AB = DE$
4. d) $AB = DE$ , $BC = EF$ , $\angle A = \angle D$
Answer: $AB = DE$ , $\angle B = \angle E$ , $BC = EF$
$AB$ and $BC$ meet at $B$ , and $DE$ and $EF$ meet at $E$ , so $\angle B = \angle E$ is the included angle. The second choice is SSA, the third is ASA and the fourth is SSS.
3.
$\triangle ABC$ and $\triangle DEF$ have $AB = DE = 5$ cm, $\angle B = \angle E = 90°$ and $BC = EF = 12$ cm. In $\triangle ABC$ , $AC = 13$ cm. How long is $DF$ , in cm?
Answer: 13
The right angles are between the equal sides, so $\triangle ABC \cong \triangle DEF$ by SAS. $DF$ corresponds to $AC$ , so $DF = 13$ cm.
4.
Segments $AE$ and $BD$ cross at $C$ . $\angle ACB = 48°$ . What is $\angle DCE$ , in degrees?
Answer: 48
$\angle ACB$ and $\angle DCE$ are vertical angles, so they are equal: $\angle DCE = 48°$ . This is the hidden angle that SAS proofs with crossing segments use.
5.
Two triangles have two pairs of equal sides and one pair of equal angles, but the angle is not between the equal sides. What can you conclude?
1. a)Congruent by ASA
2. b)Nothing for sure: this is SSA
3. c)Congruent by SAS
4. d)Congruent by SSS
Answer: Nothing for sure: this is SSA
This is SSA, which is not a congruence criterion: the same data can give two different triangles. Nothing is certain.
6.
In $\triangle PQR$ and $\triangle XYZ$ : $PQ = XY$ , $\angle Q = \angle Y$ and $\angle R = \angle Z$ . Which criterion proves $\triangle PQR \cong \triangle XYZ$ ?
1. a)SSS
2. b)ASA
3. c)AAS
4. d)SAS
Answer: AAS
The side between $\angle Q$ and $\angle R$ would be $QR$ , but the known side is $PQ$ . Two angles and a non-included side: AAS.
7.
In $\triangle ABC$ and $\triangle DEF$ : $AB = DE$ , $AC = DF$ and $\angle A = \angle D$ . Which statement is correct?
1. a) $\triangle ABC \cong \triangle DEF$ by SAS
2. b) $\triangle ABC \cong \triangle EDF$ by SAS
3. c) $\triangle ABC \cong \triangle DFE$ by SAS
4. d)Not enough information: this is SSA
Answer: $\triangle ABC \cong \triangle DEF$ by SAS
$AB$ and $AC$ meet at $A$ , so $\angle A$ is included: this is SAS, not SSA. Matching $A \leftrightarrow D$ , $B \leftrightarrow E$ (from $AB = DE$ ) and $C \leftrightarrow F$ gives $\triangle ABC \cong \triangle DEF$ .
8.
In quadrilateral $ABCD$ , $AB = CB$ and $\angle ABD = \angle CBD$ . If $\angle BAD = 95°$ , what is $\angle BCD$ , in degrees?
Answer: 95
$AB = CB$ , $\angle ABD = \angle CBD$ and the shared side $BD$ give $\triangle ABD \cong \triangle CBD$ by SAS. $\angle BCD$ corresponds to $\angle BAD$ , so it is $95°$ .
9.
Segments $AE$ and $BD$ cross at $C$ , the midpoint of both. $AB = 3x + 1$ and $ED = 2x + 6$ (in cm). Find $x$ .
Answer: 5
$AC = CE$ , $BC = CD$ and the vertical angles at $C$ give $\triangle ACB \cong \triangle ECD$ by SAS, so $AB = ED$ : $3x + 1 = 2x + 6$ , so $x = 5$ (both sides are then 16 cm).
10.
How many different triangles $ABC$ have $\angle A = 30°$ , $AB = 12$ and $BC = 8$ ?
Answer: 2
The height is $12 \div 2 = 6$ . Since $6 < 8 < 12$ , side $BC$ reaches line $AC$ at two points: 2 triangles. This is why SSA is not a criterion.
11.
$\triangle ABC$ : $AB = BC = 7$ cm and $\angle B = 40°$ . $\triangle DEF$ : $DE = EF = 7$ cm and $\angle E = 40°$ . Find $\angle D$ .
Answer: 70
1. At which vertex is the angle between sides AB and BC? B
2. Which criterion proves the triangles congruent? SAS
3. Since AB = BC, the angles at A and C are equal. What is ∠A, in degrees? 70
4. ∠D corresponds to ∠A. What is ∠D, in degrees? 70
12.
Segments $AE$ and $BD$ cross at $C$ , with $AC = CE = 6$ cm and $BC = CD = 4$ cm. If $AB = 8$ cm, how long is $DE$ ?
Answer: 8
1. Are ∠ACB and ∠ECD equal? yes
2. Which criterion proves △ACB ≅ △ECD? SAS
3. How long is DE, in cm? 8
13.
In quadrilateral $ABCD$ : $AB = AD$ , $\angle BAC = \angle DAC = 32°$ and $\angle ABC = 118°$ . Find $\angle BCD$ .
Answer: 60
1. Which side do △ABC and △ADC share? AC
2. What is ∠ACB, in degrees? 30
3. △ABC ≅ △ADC by SAS. What is ∠ACD, in degrees? 30
4. What is ∠BCD, in degrees? 60
14.
$\triangle ABC$ and $\triangle DEF$ have $AB = DE = 6$ cm, $\angle B = \angle E = 90°$ and $BC = EF = 8$ cm. Find the perimeter of $\triangle DEF$ .
Answer: 24
1. Which criterion proves △ABC ≅ △DEF? SAS
2. How long is AC, in cm? 10
3. How long is DF, in cm? 10
4. What is the perimeter of △DEF, in cm? 24
15.
$\triangle PQR$ and $\triangle STU$ have $PQ = ST = 10$ and $\angle Q = \angle T = 55°$ . Also $QR = 2y + 3$ and $TU = 15$ . For which $y$ are the triangles congruent by SAS?
Answer: 6
1. Which side of △PQR must be equal to TU? QR
2. What must QR be equal to? 15
3. Solve 2y + 3 = 15. What is y? 6
16.
Two different triangles have $\angle A = 30°$ , $AB = 8$ and $BC = 5$ . How much longer is $AC$ in the larger triangle than in the smaller one?
Answer: 6
1. What is the height of B above line AC? 4
2. How many different triangles are possible? 2
3. How far is each possible C from the foot of the height? 3
4. How much longer is the larger AC than the smaller one? 6
17.
The diagonals $AC$ and $BD$ of quadrilateral $ABCD$ cross at $M$ , the midpoint of both. $AM = 4.5$ cm, $AB = 8$ cm and $BC = 5$ cm. Find the perimeter of $ABCD$ .
Answer: 26
1. How long is CM, in cm? 4.5
2. Which criterion proves △AMB ≅ △CMD? SAS
3. How long is CD, in cm? 8
4. In the same way △BMC ≅ △DMA. How long is AD, in cm? 5
5. What is the perimeter of ABCD, in cm? 26
18.
Four pairs of triangles are described below. For each pair, decide whether the triangles are guaranteed to be congruent, then count how many pairs are.
Answer: 2
1. Pair 1: two sides and the angle between them are equal. Congruent for sure? yes
2. Pair 2: two sides and an angle that is not between them are equal. Congruent for sure? no
3. Pair 3: all three angles are equal. Congruent for sure? no
4. Pair 4: two angles and a side that is not between them are equal. Congruent for sure? yes
5. How many of the four pairs are congruent for sure? 2