Practice
Angles of a Triangle Practice
Fifty original questions on triangle sums, exterior angles, isosceles relationships, and chained theorem reasoning.
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Question 1
Explanation
The Angle Sum Theorem gives \(42+63+m\angle C=180\), so \(m\angle C=75^\circ\).
- Reasoning
Add the two known interior angles and subtract from 180 degrees.
Question 2
Explanation
The Angle Sum Theorem gives \(28+91+m\angle C=180\), so \(m\angle C=61^\circ\).
- Reasoning
Add the two known interior angles and subtract from 180 degrees.
Question 3
Explanation
The Angle Sum Theorem gives \(57+74+m\angle C=180\), so \(m\angle C=49^\circ\).
- Reasoning
Add the two known interior angles and subtract from 180 degrees.
Question 4
Explanation
The Angle Sum Theorem gives \(36+88+m\angle C=180\), so \(m\angle C=56^\circ\).
- Reasoning
Add the two known interior angles and subtract from 180 degrees.
Question 5
Explanation
The Angle Sum Theorem gives \(69+52+m\angle C=180\), so \(m\angle C=59^\circ\).
- Reasoning
Add the two known interior angles and subtract from 180 degrees.
Question 6
Explanation
The Angle Sum Theorem gives \(47+79+m\angle C=180\), so \(m\angle C=54^\circ\).
- Reasoning
Add the two known interior angles and subtract from 180 degrees.
Question 7
Explanation
The Angle Sum Theorem gives \(81+34+m\angle C=180\), so \(m\angle C=65^\circ\).
- Reasoning
Add the two known interior angles and subtract from 180 degrees.
Question 8
Explanation
The Angle Sum Theorem gives \(25+116+m\angle C=180\), so \(m\angle C=39^\circ\).
- Reasoning
Add the two known interior angles and subtract from 180 degrees.
Question 9
Explanation
Angle sum gives \(5x+30=180\), so \(x=30\). Substitution gives the first angle \(70^\circ\).
- Reasoning
Set all three expressions equal to 180, solve x, then evaluate the requested angle.
Question 10
Explanation
Angle sum gives \(5x+30=180\), so \(x=30\). Substitution gives the first angle \(96^\circ\).
- Reasoning
Set all three expressions equal to 180, solve x, then evaluate the requested angle.
Question 11
Explanation
Angle sum gives \(6x+24=180\), so \(x=26\). Substitution gives the first angle \(112^\circ\).
- Reasoning
Set all three expressions equal to 180, solve x, then evaluate the requested angle.
Question 12
Explanation
Angle sum gives \(7x+40=180\), so \(x=20\). Substitution gives the first angle \(105^\circ\).
- Reasoning
Set all three expressions equal to 180, solve x, then evaluate the requested angle.
Question 13
Explanation
Angle sum gives \(6x+30=180\), so \(x=25\). Substitution gives the first angle \(59^\circ\).
- Reasoning
Set all three expressions equal to 180, solve x, then evaluate the requested angle.
Question 14
Explanation
Angle sum gives \(8x+36=180\), so \(x=18\). Substitution gives the first angle \(66^\circ\).
- Reasoning
Set all three expressions equal to 180, solve x, then evaluate the requested angle.
Question 15
Explanation
Angle sum gives \(7x+40=180\), so \(x=20\). Substitution gives the first angle \(84^\circ\).
- Reasoning
Set all three expressions equal to 180, solve x, then evaluate the requested angle.
Question 16
Explanation
By the Exterior Angle Theorem, the exterior measure is \(31+58=89^\circ\).
- Reasoning
Add the two remote interior angles, not the adjacent interior angle.
Question 17
Explanation
By the Exterior Angle Theorem, the exterior measure is \(44+73=117^\circ\).
- Reasoning
Add the two remote interior angles, not the adjacent interior angle.
Question 18
Explanation
By the Exterior Angle Theorem, the exterior measure is \(27+96=123^\circ\).
- Reasoning
Add the two remote interior angles, not the adjacent interior angle.
Question 19
Explanation
By the Exterior Angle Theorem, the exterior measure is \(62+49=111^\circ\).
- Reasoning
Add the two remote interior angles, not the adjacent interior angle.
Question 20
Explanation
By the Exterior Angle Theorem, the exterior measure is \(38+77=115^\circ\).
- Reasoning
Add the two remote interior angles, not the adjacent interior angle.
Question 21
Explanation
By the Exterior Angle Theorem, the exterior measure is \(53+68=121^\circ\).
- Reasoning
Add the two remote interior angles, not the adjacent interior angle.
Question 22
Explanation
By the Exterior Angle Theorem, the exterior measure is \(72+35=107^\circ\).
- Reasoning
Add the two remote interior angles, not the adjacent interior angle.
Question 23
Explanation
By the Exterior Angle Theorem, the exterior measure is \(46+89=135^\circ\).
- Reasoning
Add the two remote interior angles, not the adjacent interior angle.
Question 24
Explanation
A linear pair sums to 180 degrees, so the exterior angle is \(180-43=137^\circ\).
- Reasoning
Use the straight-angle relationship when the adjacent interior angle is known.
Question 25
Explanation
A linear pair sums to 180 degrees, so the exterior angle is \(180-56=124^\circ\).
- Reasoning
Use the straight-angle relationship when the adjacent interior angle is known.
Question 26
Explanation
A linear pair sums to 180 degrees, so the exterior angle is \(180-72=108^\circ\).
- Reasoning
Use the straight-angle relationship when the adjacent interior angle is known.
Question 27
Explanation
A linear pair sums to 180 degrees, so the exterior angle is \(180-104=76^\circ\).
- Reasoning
Use the straight-angle relationship when the adjacent interior angle is known.
Question 28
Explanation
A linear pair sums to 180 degrees, so the exterior angle is \(180-119=61^\circ\).
- Reasoning
Use the straight-angle relationship when the adjacent interior angle is known.
Question 29
Explanation
Equal sides AB and BC make opposite base angles A and C equal. Each is \((180-104)/2=38^\circ\).
- Reasoning
Map equal sides to their opposite angles, then split the remaining angle sum equally.
Question 30
Explanation
Equal sides AB and BC make opposite base angles A and C equal. Each is \((180-78)/2=51^\circ\).
- Reasoning
Map equal sides to their opposite angles, then split the remaining angle sum equally.
Question 31
Explanation
Equal sides AB and BC make opposite base angles A and C equal. Each is \((180-52)/2=64^\circ\).
- Reasoning
Map equal sides to their opposite angles, then split the remaining angle sum equally.
Question 32
Explanation
Equal sides AB and BC make opposite base angles A and C equal. Each is \((180-36)/2=72^\circ\).
- Reasoning
Map equal sides to their opposite angles, then split the remaining angle sum equally.
Question 33
Explanation
Equal sides AB and BC make opposite base angles A and C equal. Each is \((180-86)/2=47^\circ\).
- Reasoning
Map equal sides to their opposite angles, then split the remaining angle sum equally.
Question 34
Explanation
Equal sides AB and BC make opposite base angles A and C equal. Each is \((180-62)/2=59^\circ\).
- Reasoning
Map equal sides to their opposite angles, then split the remaining angle sum equally.
Question 35
Explanation
Equal sides AB and BC make opposite base angles A and C equal. Each is \((180-114)/2=33^\circ\).
- Reasoning
Map equal sides to their opposite angles, then split the remaining angle sum equally.
Question 36
Explanation
Equal sides AB and BC make opposite base angles A and C equal. Each is \((180-44)/2=68^\circ\).
- Reasoning
Map equal sides to their opposite angles, then split the remaining angle sum equally.
Question 37
Explanation
Congruent angles have congruent opposite sides. The side opposite angle B is AC, and the side opposite angle C is AB.
- Reasoning
Trace from each marked angle to the side directly opposite it.
Question 38
Explanation
Congruent angles have congruent opposite sides. The side opposite angle A is BC, and the side opposite angle C is AB.
- Reasoning
Trace from each marked angle to the side directly opposite it.
Question 39
Explanation
Congruent angles have congruent opposite sides. The side opposite angle A is BC, and the side opposite angle B is AC.
- Reasoning
Trace from each marked angle to the side directly opposite it.
Question 40
Explanation
Congruent angles have congruent opposite sides. The side opposite angle B is AC, and the side opposite angle C is AB.
- Reasoning
Trace from each marked angle to the side directly opposite it.
Question 41
Explanation
Congruent angles have congruent opposite sides. The side opposite angle A is BC, and the side opposite angle C is AB.
- Reasoning
Trace from each marked angle to the side directly opposite it.
Question 42
Explanation
Under the isosceles vertex-bisector conditions, BD bisects the base, so \(DC=AD=18\) and \(AC=36\).
- Reasoning
Use the corollary only after confirming the triangle is isosceles and the vertex angle is bisected.
Question 43
Explanation
Under the isosceles vertex-bisector conditions, BD bisects the base, so \(DC=AD=23\) and \(AC=46\).
- Reasoning
Use the corollary only after confirming the triangle is isosceles and the vertex angle is bisected.
Question 44
Explanation
Under the isosceles vertex-bisector conditions, BD bisects the base, so \(DC=AD=31\) and \(AC=62\).
- Reasoning
Use the corollary only after confirming the triangle is isosceles and the vertex angle is bisected.
Question 45
Explanation
Under the isosceles vertex-bisector conditions, BD bisects the base, so \(DC=AD=37\) and \(AC=74\).
- Reasoning
Use the corollary only after confirming the triangle is isosceles and the vertex angle is bisected.
Question 46
Explanation
The acute angles of a right triangle are complementary: \(3x+6+5x+4=90\), so \(x=10\) and the smaller angle is 36 degrees.
- Reasoning
Set the two acute angles equal to 90, not 180.
Question 47
Explanation
The adjacent base angle is \(180-128=52^\circ\). Both base angles are 52 degrees, so the vertex angle is \(180-104=76^\circ\).
- Reasoning
Use the linear pair, then the isosceles theorem, then angle sum.
Question 48
Explanation
The exterior and adjacent interior form a linear pair totaling 180 degrees, while the Exterior Angle Theorem uses the two remote interiors.
- Reasoning
Distinguish the linear-pair relationship from the remote-angle theorem.
Question 49
Explanation
Geometry conclusions come from stated facts and markings, not visual appearance, especially when a figure may not be drawn to scale.
- Reasoning
Require explicit evidence before applying the isosceles theorem.
Question 50
Explanation
The Exterior Angle Theorem gives \(2x+9+3x-4=110\). Thus \(5x+5=110\) and \(x=21\).
- Reasoning
Add the remote expressions and set their sum equal to the exterior angle.
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Questions to review
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- Question 1Triangle angle sumEasy
- Question 2Triangle angle sumEasy
- Question 3Triangle angle sumEasy
- Question 4Triangle angle sumEasy
- Question 5Triangle angle sumEasy
- Question 6Triangle angle sumEasy
- Question 7Triangle angle sumEasy
- Question 8Triangle angle sumEasy
- Question 9Algebraic triangle anglesEasy
- Question 10Algebraic triangle anglesEasy
- Question 11Algebraic triangle anglesEasy
- Question 12Algebraic triangle anglesEasy
- Question 13Algebraic triangle anglesEasy
- Question 14Algebraic triangle anglesEasy
- Question 15Algebraic triangle anglesEasy
- Question 16Exterior Angle TheoremMedium
- Question 17Exterior Angle TheoremMedium
- Question 18Exterior Angle TheoremMedium
- Question 19Exterior Angle TheoremMedium
- Question 20Exterior Angle TheoremMedium
- Question 21Exterior Angle TheoremMedium
- Question 22Exterior Angle TheoremMedium
- Question 23Exterior Angle TheoremMedium
- Question 24Exterior linear pairMedium
- Question 25Exterior linear pairMedium
- Question 26Exterior linear pairMedium
- Question 27Exterior linear pairMedium
- Question 28Exterior linear pairMedium
- Question 29Isosceles Triangle TheoremMedium
- Question 30Isosceles Triangle TheoremMedium
- Question 31Isosceles Triangle TheoremMedium
- Question 32Isosceles Triangle TheoremMedium
- Question 33Isosceles Triangle TheoremMedium
- Question 34Isosceles Triangle TheoremMedium
- Question 35Isosceles Triangle TheoremMedium
- Question 36Isosceles Triangle TheoremMedium
- Question 37Converse Isosceles TheoremMedium
- Question 38Converse Isosceles TheoremMedium
- Question 39Converse Isosceles TheoremMedium
- Question 40Converse Isosceles TheoremMedium
- Question 41Converse Isosceles TheoremHard
- Question 42Isosceles vertex-bisector corollaryHard
- Question 43Isosceles vertex-bisector corollaryHard
- Question 44Isosceles vertex-bisector corollaryHard
- Question 45Isosceles vertex-bisector corollaryHard
- Question 46Right-triangle complementary anglesHard
- Question 47Chained triangle theoremsHard
- Question 48Error analysisHard
- Question 49Diagram interpretationHard
- Question 50Exterior-angle algebraHard