If m∠2 = 98°, m∠3 = 23°, and m∠8 = 70°, find each measure.
Understand the Problem
The question is asking to find the measures of various angles in a triangle given the measures of three angles. It involves using geometric principles, particularly the properties of angles in triangles, to calculate the unknown angles based on the known measures.
Answer
a. $m\angle 1 = 59^\circ$, b. $m\angle 4 = 98^\circ$, c. $m\angle 5 = 59^\circ$, d. $m\angle 6 = 110^\circ$, e. $m\angle 7 = 157^\circ$, f. $m\angle 9 = 70^\circ$.
Answer for screen readers
a. $m\angle 1 = 59^\circ$
b. $m\angle 4 = 98^\circ$
c. $m\angle 5 = 59^\circ$
d. $m\angle 6 = 110^\circ$
e. $m\angle 7 = 157^\circ$
f. $m\angle 9 = 70^\circ$
Steps to Solve
- Identify known angles The given angles are:
- $m\angle 2 = 98^\circ$
- $m\angle 3 = 23^\circ$
- $m\angle 8 = 70^\circ$
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Find the angle sum of triangle The sum of the angles in any triangle is $180^\circ$. To find $m\angle 1$, which is part of triangle containing angles 2 and 3, we use: $$ m\angle 1 = 180^\circ - (m\angle 2 + m\angle 3) $$
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Calculate angle $m\angle 1$ Substituting the known values: $$ m\angle 1 = 180^\circ - (98^\circ + 23^\circ) $$ $$ m\angle 1 = 180^\circ - 121^\circ = 59^\circ $$
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Find angles 4 and 5 Angles 4 and 5 are vertical angles to angles 2 and 1 respectively. Thus:
- $m\angle 4 = m\angle 2 = 98^\circ$
- $m\angle 5 = m\angle 1 = 59^\circ$
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Find angle 6 To find angle 6, we check the remaining angles in the triangle, where angle 6 is supplementary to angle 8: $$ m\angle 6 = 180^\circ - m\angle 8 $$ $$ m\angle 6 = 180^\circ - 70^\circ = 110^\circ $$
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Find angle 7 Angle 7 is directly related to angle 3. Since angle 7 and angle 3 are adjacent interior angles on the same side of the transversal, we find: $$ m\angle 7 = 180^\circ - m\angle 3 $$ $$ m\angle 7 = 180^\circ - 23^\circ = 157^\circ $$
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Find angle 9 Angle 9 is supplementary to angle 6: $$ m\angle 9 = 180^\circ - m\angle 6 $$ $$ m\angle 9 = 180^\circ - 110^\circ = 70^\circ $$
a. $m\angle 1 = 59^\circ$
b. $m\angle 4 = 98^\circ$
c. $m\angle 5 = 59^\circ$
d. $m\angle 6 = 110^\circ$
e. $m\angle 7 = 157^\circ$
f. $m\angle 9 = 70^\circ$
More Information
Each angle's measure was calculated using properties of triangles and supplementary angles. This ensures all angles are correctly derived based on the known values.
Tips
- Confusing vertical angles, which are equal, with adjacent angles, which may not be.
- Failing to accurately apply the properties of triangle angle sums.
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