12 multiple-choice questions, progressively harder.
A triangle has side lengths 9,9,139,9,139,9,13 centimeters. Which side classification applies?
Solution
Correct answer: C
Two sides have the same length, so they are congruent. The triangle is isosceles because it has at least two congruent sides.
The third side has a different length, so it is not equilateral.
A triangle has interior angles 42∘,63∘,75∘42^\circ,63^\circ,75^\circ42∘,63∘,75∘. How is it classified by angles?
Correct answer: A
Every interior angle is less than 90∘90^\circ90∘, so the triangle is acute.
The measures also fit the triangle total: 42∘+63∘+75∘=180∘.42^\circ+63^\circ+75^\circ=180^\circ.42∘+63∘+75∘=180∘. Scalene is a side classification, which these angle definitions do not determine.
In triangle ABCABCABC, m∠CAB=61∘m\angle CAB=61^\circm∠CAB=61∘ and m∠ABC=74∘m\angle ABC=74^\circm∠ABC=74∘. Find m∠ACBm\angle ACBm∠ACB.
Correct answer: D
The three interior angle measures total 180∘180^\circ180∘. Subtract the two known measures.
m∠ACB=180∘−61∘−74∘=45∘.m\angle ACB=180^\circ-61^\circ-74^\circ=45^\circ.m∠ACB=180∘−61∘−74∘=45∘.
Triangle PQRPQRPQR is right at QQQ, and m∠QPR=27∘m\angle QPR=27^\circm∠QPR=27∘. What is m∠PRQm\angle PRQm∠PRQ?
Correct answer: B
The two acute angles of a right triangle are complementary because the right angle uses 90∘90^\circ90∘ of the 180∘180^\circ180∘ total.
m∠PRQ=90∘−27∘=63∘.m\angle PRQ=90^\circ-27^\circ=63^\circ.m∠PRQ=90∘−27∘=63∘.
Each of two triangles has one angle of 84∘84^\circ84∘ and another of 39∘39^\circ39∘. What does the third-angle theorem give?
Subtract the same two measures from 180∘180^\circ180∘ in each triangle.
180∘−84∘−39∘=57∘.180^\circ-84^\circ-39^\circ=57^\circ.180∘−84∘−39∘=57∘.
Equal angle measures give congruent angles. The theorem supplies no side-length equality.
A triangle has an interior angle of 102∘102^\circ102∘. Which statement about its other two interior angles must be true?
The remaining two positive measures total 180∘−102∘=78∘.180^\circ-102^\circ=78^\circ.180∘−102∘=78∘.
Each is smaller than their positive total, so each is less than 90∘90^\circ90∘ and therefore acute.
Which extra line is used in the triangle angle-sum proof?
The auxiliary parallel through a vertex creates alternate interior angles congruent to the other two triangle angles.
Those two angles and the original angle at the vertex form adjacent parts of one straight angle, with total 180 degrees.
The three interior angles of a triangle measure (x+5)∘(x+5)^\circ(x+5)∘, (2x+10)∘(2x+10)^\circ(2x+10)∘, and (3x+15)∘(3x+15)^\circ(3x+15)∘. Find xxx.
The triangle angle sum supplies the equation.
(x+5)+(2x+10)+(3x+15)=180.(x+5)+(2x+10)+(3x+15)=180.(x+5)+(2x+10)+(3x+15)=180.
Combine terms to obtain 6x+30=1806x+30=1806x+30=180, then subtract 303030 and divide by 666.
x=25.x=25.x=25.
The angle measures become 30∘,60∘,90∘30^\circ,60^\circ,90^\circ30∘,60∘,90∘, which total 180∘180^\circ180∘.
An obtuse triangle has angle measures 112∘112^\circ112∘ and 28∘28^\circ28∘ at two vertices. What is the third measure?
The known measures total 140∘140^\circ140∘. The third measure is the part left in the triangle total.
180∘−112∘−28∘=40∘.180^\circ-112^\circ-28^\circ=40^\circ.180∘−112∘−28∘=40∘.
Two triangles have two pairs of congruent angles. Which conclusion follows directly from the third-angle theorem?
Each remaining angle is found by subtracting the same two known angle measures from 180 degrees.
The remaining measures are equal, so those angles are congruent. Neither side lengths nor a right angle were supplied.
Under the inclusive side definitions, which statement is true?
An equilateral triangle has three congruent sides. It therefore has at least two congruent sides, satisfying the isosceles definition.
The isosceles definition does not require the third side to be congruent to the other two.
What is the largest possible number of interior angles measuring at least 90∘90^\circ90∘ in a nondegenerate triangle?
Two such angles would already total at least 180∘180^\circ180∘. The third angle must be positive, contradicting the triangle sum.
One is possible, for example with angles 90∘,41∘,49∘90^\circ,41^\circ,49^\circ90∘,41∘,49∘, so the maximum is one.
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