Geometric Proofs

Show your work for each problem.

1.
Lines $AB$ and $CD$ are parallel ( $A$ and $C$ on the left, $B$ and $D$ on the right). A transversal crosses $AB$ at $E$ and $CD$ at $F$ . $\angle AEF = 72^\circ$ and $\angle EFD = 72^\circ$ . Which reason justifies $\angle EFD = \angle AEF$ ?
1. a)Same-side interior angles are equal
2. b)Vertical angles are equal
3. c)Alternate interior angles are equal
4. d)Corresponding angles are equal
Answer: Alternate interior angles are equal
$\angle AEF$ (at $E$ , on the left) and $\angle EFD$ (at $F$ , on the right) lie between the parallel lines on opposite sides of the transversal: they are alternate interior angles, which are equal at parallel lines.
2.
Lines $AB$ and $CD$ are parallel ( $A$ and $C$ on the left, $B$ and $D$ on the right). A transversal crosses $AB$ at $E$ and $CD$ at $F$ . $\angle AEF = 72^\circ$ . How many degrees is $\angle CFE$ ?
Answer: 108
$\angle EFD = \angle AEF = 72^\circ$ (alternate interior angles). $\angle CFE$ and $\angle EFD$ form a straight angle, so $\angle CFE = 180^\circ - 72^\circ = 108^\circ$ . Same-side interior check: $72^\circ + 108^\circ = 180^\circ$ .
3.
In $\triangle ABC$ , $\angle A = 47^\circ$ and $\angle B = 68^\circ$ . How many degrees is $\angle C$ ?
Answer: 65
By the triangle angle sum theorem, $\angle C = 180^\circ - 47^\circ - 68^\circ = 65^\circ$ .
4.
Which set of equal parts does NOT prove that two triangles are congruent?
1. a)SAS
2. b)AAA
3. c)SSS
4. d)ASA
Answer: AAA
Three equal angles (AAA) fix only the shape, not the size: a small and a large equilateral triangle both have three $60^\circ$ angles but are not congruent. SSS, SAS and ASA are valid criteria.
5.
In $\triangle ABC$ , $AB = AC$ and $\angle B = 52^\circ$ . Find $\angle C$ and $\angle A$ .
Answer: 76
1. How many degrees is $\angle C$ ? 52
2. How many degrees is $\angle B + \angle C$ ? 104
3. How many degrees is $\angle A$ ? 76
6.
Parallel lines $\ell$ and $m$ ( $\ell$ above $m$ ) are cut by a transversal $t$ . At $\ell$ , the angle above $\ell$ and to the right of $t$ is $110^\circ$ . Find the angle above $m$ to the right of $t$ , then the angle above $m$ to the left of $t$ .
Answer: 70
1. How many degrees is the angle above $m$ to the right of $t$ ? 110
2. How many degrees is the angle above $m$ to the left of $t$ ? 70
7.
Segments $AD$ and $BC$ cross at $M$ , and $M$ is the midpoint of both. You know $AM = DM$ , $BM = CM$ and $\angle AMB = \angle DMC$ (vertical angles). Which criterion proves $\triangle AMB \cong \triangle DMC$ ?
1. a)AAS
2. b)ASA
3. c)SAS
4. d)SSS
Answer: SAS
Two sides ( $AM$ , $BM$ ) and the angle between them (at $M$ ) match two sides and the angle between them in the other triangle: that is SAS.
8.
In $\triangle ABC$ , the line through $A$ parallel to $BC$ makes an angle of $55^\circ$ with $AB$ (on the side of $B$ ) and an angle of $64^\circ$ with $AC$ (on the side of $C$ ). How many degrees is $\angle BAC$ ?
Answer: 61
The three angles at $A$ lie on the parallel line, so $55^\circ + \angle BAC + 64^\circ = 180^\circ$ and $\angle BAC = 61^\circ$ . By alternate interior angles, $\angle B = 55^\circ$ and $\angle C = 64^\circ$ , and indeed $55^\circ + 61^\circ + 64^\circ = 180^\circ$ .
9.
In $\triangle ABC$ , $AB = AC$ and $\angle A = 36^\circ$ . Find the base angle $\angle B$ , then the exterior angle at $B$ (between $AB$ and the extension of $CB$ beyond $B$ ).
Answer: 108
1. How many degrees is $\angle B + \angle C$ ? 144
2. How many degrees is $\angle B$ ? 72
3. How many degrees is the exterior angle at $B$ ? 108
10.
Segments $AE$ and $BD$ cross at $C$ , $AB \parallel DE$ and $C$ is the midpoint of $AE$ . Also $\angle BAC = 40^\circ$ , $\angle ACB = 75^\circ$ and $AB = 9$ cm. Find the angles of $\triangle EDC$ and the length $DE$ .
Answer: 9
1. How many degrees is $\angle CED$ ? 40
2. How many degrees is $\angle ECD$ ? 75
3. How many degrees is $\angle CDE$ ? 65
4. How many cm is $DE$ ? 9
11.
In $\triangle ABC$ and $\triangle DEF$ : $AB = DE$ , $\angle A = \angle D$ and $\angle B = \angle E$ . Which criterion proves $\triangle ABC \cong \triangle DEF$ ?
1. a)SSS
2. b)AAS
3. c)SAS
4. d)ASA
Answer: ASA
The equal side $AB$ joins the vertices of the two equal angles, so it lies between them: angle, side, angle, that is ASA.
12.
$AD$ bisects $\angle BAC$ and $\angle BAD = 34^\circ$ . How many degrees is $\angle BAC$ ?
Answer: 68
By the definition of angle bisector, $\angle CAD = \angle BAD = 34^\circ$ , so $\angle BAC = 34^\circ + 34^\circ = 68^\circ$ .
13.
Two parallel lines are cut by a transversal. A pair of alternate interior angles measures $(3x + 10)^\circ$ and $(5x - 30)^\circ$ . Find $x$ , the angle, and the angle that forms a linear pair with it.
Answer: 110
1. Solve $3x + 10 = 5x - 30$ . What is $x$ ? 20
2. How many degrees is each alternate interior angle? 70
3. How many degrees is the angle that forms a linear pair with it? 110
14.
In $\triangle ABC$ , $AB = AC$ , $\angle B = (2x + 10)^\circ$ and $\angle C = (3x - 15)^\circ$ . How many degrees is $\angle A$ ?
Answer: 60
$2x + 10 = 3x - 15$ gives $x = 25$ , so $\angle B = \angle C = 60^\circ$ . Then $\angle A = 180^\circ - 60^\circ - 60^\circ = 60^\circ$ : the triangle is even equilateral.
15.
In quadrilateral $ABCD$ , $AB = AD$ and $CB = CD$ , with $\angle BAD = 80^\circ$ and $\angle BCD = 60^\circ$ . Use $\triangle ABC \cong \triangle ADC$ (SSS) to find $\angle ABC$ and $\angle ADC$ .
Answer: 110
1. How many degrees is $\angle BAC$ ? 40
2. How many degrees is $\angle BCA$ ? 30
3. How many degrees is $\angle ABC$ ? 110
4. How many degrees is $\angle ADC$ ? 110
16.
Proof that the base angles of an isosceles triangle are equal. Given: $AB = AC$ , and $AD$ bisects $\angle BAC$ with $D$ on $BC$ . Step 1: $AB = AC$ (given). Step 2: $\angle BAD = \angle CAD$ (definition of angle bisector). Step 3: $AD = AD$ . Step 4: $\triangle ABD \cong \triangle ACD$ (SAS). Step 5: $\angle B = \angle C$ (CPCTC). What is the reason for step 3?
1. a)Definition of midpoint
2. b)Reflexive property
3. c)Transitive property
4. d)Vertical angles are equal
Answer: Reflexive property
$AD$ is a shared side of $\triangle ABD$ and $\triangle ACD$ . Every segment is equal to itself: this is the reflexive property. It gives the second pair of sides needed for SAS.
17.
In $\triangle ABC$ , $\angle A = 50^\circ$ and $\angle B = 60^\circ$ . Side $BC$ is extended beyond $C$ to a point $D$ . Find $\angle ACB$ , the exterior angle $\angle ACD$ , and compare it with $\angle A + \angle B$ .
Answer: 110
1. How many degrees is $\angle ACB$ ? 70
2. How many degrees is $\angle ACD$ ? 110
3. How many degrees is $\angle A + \angle B$ ? 110
18.
In $\triangle ABC$ , $AB = AC$ and $\angle A = 44^\circ$ . A segment $DE$ with $D$ on $AB$ and $E$ on $AC$ is parallel to $BC$ . Find $\angle ABC$ , $\angle ADE$ and $\angle BDE$ .
Answer: 112
1. How many degrees is $\angle ABC$ ? 68
2. How many degrees is $\angle ADE$ ? 68
3. How many degrees is $\angle BDE$ ? 112