1.6/#3
Given: diagram, m< 𝑂𝑃𝑇 = 90
Prove: the measure of < 𝑉𝐴𝑌 is twice
that of < 𝑅𝑃𝑇
According to the diagram and the given,m ∠OPR +m ∠RPT = 90. Also from the diagram
we know that m∠OPT = 4x and m∠RPT = x. If we substitute that into the original
equation we get:
4x + x = 90
5x = 90
x = 18
From the diagram we see that m∠VAY = x + 18. Substitute x = 18 into this equation and
we get m∠VAY = x + 18 = 18 + 18 = 36 and m∠RPT = x = 18.
36 = 2 * 18 therefore m∠VAY is twice the m∠RPT.
1.6/#6
Given: ∠1 is obtuse
∠2 is acute
Prove: < 1 ≅< 2
This conclusion is not true. Since ∠1 is obtuse, then 90<m∠1<180, because if an angle
is obtuse then its measure is between 90 and 180. Since ∠2 is acute, then 0<m∠1<90,
because if an angle is acute then its measure is between 0 and 90. These measures
have no points of intersection, therefore they cannot be ≅ .
1.6/#7
Prove that if < 1 ≅< 2, they are both right angles.
From the diagram we can assume that a straight angle is formed. If an angle is a
straight angle, then its measure is 180 ∴ m∠ABC = 180. We can also see from the
diagram that m∠1 + m∠2 = m∠ABC ∴ m∠1 + m∠2 = 180
From the given we know that ∠1 and ∠2 are congruent, ∴ m∠1=m∠2=x, because if 2
angles are congruent, then they have equal measure. Substituting the x into the above
equation we get:
x + x = 180
2x = 180
x = 90
Since m∠1=m∠2=x=90, we know that both ∠1 and ∠2 are right angles, because if the
measure of an angle is 90, then it is a right angle.
1.6/#8
If an obtuse angle is bisected, each of the two resulting angles is acute.
If an angle is obtuse, then its measure is between 90 and 180. This can be written as:
90<m<180
If a ray bisects an angle, then it is divided into two congruent angles. If two angles are
congruent, then they have equal measure ∴ m∠1=m∠2=x. Substituting m∠1 + m∠2 in
for the m in the above inequality gets us:
90< m∠1 + m∠2 <180
90< x + x <180
90
2𝑥
180
< <
2
2
2
45 < x < 90
If an angle has a measure between 0 and 90, then the angle is acute. m∠1 and m∠2
are both equal to x, which is between 45 and 90, therefore the bisected angles are
acute.
1.6/#9
Given: → bisects ∠BCD
𝐶𝐸
∠A is a right angle
m∠BCE=45
Prove: < 𝐴 ≅< 𝐵𝐶𝐷
From the given → bisects ∠BCD ∴ < 𝐵𝐶𝐸 ≅< 𝐸𝐶𝐷 because if a ray bisects and angle,
𝐶𝐸
then it divides it into 2 congruent angles. If 2 angles are congruent, then they have
equal measure ∴ m∠BCE=m∠ECD. From the given m∠BCE=45 ∴ m∠ECD also = 45.
From the diagram we can see that m∠BCE + m∠ECD = m∠BCD. Substituting into that
equation gets us: 45 + 45 = m∠BCD
90 = m∠BCD
If the measure of an angle is 90, then it is a right angle ∴ ∠BCD is a right angle. We also
know from the given that ∠A is a right angle. If two angles are right angles, then they
are congruent ∴ < 𝐴 ≅< 𝐵𝐶𝐷.