Question
page 24 of 24 Angles Inside a Circle Quick Check 1 of 51 of 5 Items Question Circle C is intersected by tangent lines GH and JH . The two tangent lines intersect each other at point H . Identify the measure of ∠GHJ if m∠GCJ is 134°.(1 point) Responses 46° 46° 226° 226° 67° 67° 134° 134° Skip to navigation
Answers
GPT-4o mini
To solve this problem, we can use the fact that the angle formed between two intersecting tangent lines that intersect at a point outside the circle is equal to half the difference of the measures of the arcs they subtend.
Given:
- \( m\angle GCJ = 134^\circ \)
- Lines GH and JH are tangent to the circle at points G and J, respectively.
Since \( \angle GHJ \) is formed by the two tangents and \( \angle GCJ \) is related to the arcs subtended by those tangents, we can use the following formula to find \( m\angle GHJ \):
\[
m\angle GHJ = \frac{1}{2} | m\overarc{GJ} - m\overarc{HJ} |
\]
However, since we do not have the measures of the arcs directly, we can deduce that:
1. The angle formed by tangents \( GH \) and \( JH \) at point H is equal to half the difference between the interior angle \( \angle GCJ \) and the angle formed at the center related to arc \( GJ \).
Since \( m\angle GCJ \) is given as \( 134^\circ \), and considering the property of linear pairs, we can deduce that angle \( \angle GHJ \) will be supplementary to \( \angle GCJ \):
\[
m\angle GHJ = 180^\circ - m\angle GCJ
\]
\[
m\angle GHJ = 180^\circ - 134^\circ = 46^\circ
\]
Thus, the measure of \( \angle GHJ \) is \( 46° \).
**Final Answer: 46°**
Given:
- \( m\angle GCJ = 134^\circ \)
- Lines GH and JH are tangent to the circle at points G and J, respectively.
Since \( \angle GHJ \) is formed by the two tangents and \( \angle GCJ \) is related to the arcs subtended by those tangents, we can use the following formula to find \( m\angle GHJ \):
\[
m\angle GHJ = \frac{1}{2} | m\overarc{GJ} - m\overarc{HJ} |
\]
However, since we do not have the measures of the arcs directly, we can deduce that:
1. The angle formed by tangents \( GH \) and \( JH \) at point H is equal to half the difference between the interior angle \( \angle GCJ \) and the angle formed at the center related to arc \( GJ \).
Since \( m\angle GCJ \) is given as \( 134^\circ \), and considering the property of linear pairs, we can deduce that angle \( \angle GHJ \) will be supplementary to \( \angle GCJ \):
\[
m\angle GHJ = 180^\circ - m\angle GCJ
\]
\[
m\angle GHJ = 180^\circ - 134^\circ = 46^\circ
\]
Thus, the measure of \( \angle GHJ \) is \( 46° \).
**Final Answer: 46°**