State the theorem or postulate that is related to the measures of the angles in each pair. Then find the angle measures.
m∠3=(23x+11)°,m∠4=(14x+21)°
no idea. how are the angles related?
vertical angles are congruent
if these angles are on a transversal between parallel lines, review that topic
The theorem that is related to the measures of the angles in each pair is the Vertical Angle Theorem. According to this theorem, when two lines intersect, the pairs of vertical angles formed are congruent.
Given that m∠3 = (23x + 11)° and m∠4 = (14x + 21)°, we can set these two expressions equal to each other:
(23x + 11)° = (14x + 21)°
Now, solve for x:
23x + 11 = 14x + 21
23x - 14x = 21 - 11
9x = 10
x = 10/9
Now substitute the value of x back into the angle measures to find their values:
m∠3 = (23x + 11)° = (23(10/9) + 11)° = 33°
m∠4 = (14x + 21)° = (14(10/9) + 21)° = 35°
Therefore, m∠3 = 33° and m∠4 = 35°.
To find the angle measures of ∠3 and ∠4, we need to identify the theorem or postulate that relates these angles. From the given information, it appears that ∠3 and ∠4 may be alternate interior angles or corresponding angles. However, without further information about the context or the figure, we cannot definitively determine the relationship between these angles.
Let's assume the angles are corresponding angles. In that case, the Corresponding Angles Postulate states that if two parallel lines are intersected by a transversal, then the corresponding angles formed are congruent.
Let ∠3 and ∠4 be corresponding angles. According to the Corresponding Angles Postulate, we have:
m∠3 = m∠4
Now, let's express the given angle measures in terms of x:
m∠3 = (23x + 11)°
m∠4 = (14x + 21)°
Since the angles are congruent, we can set up an equation:
(23x + 11)° = (14x + 21)°
Now, solve for x:
23x + 11 = 14x + 21
23x - 14x = 21 - 11
9x = 10
x = 10/9
Now, substitute the value of x back into the expressions for the angles to find their measures:
m∠3 = (23x + 11)°
m∠3 = (23 * (10/9) + 11)°
m∠3 ≈ 34.22°
m∠4 = (14x + 21)°
m∠4 = (14 * (10/9) + 21)°
m∠4 ≈ 35.78°
Therefore, if ∠3 and ∠4 are corresponding angles, their measures are approximately 34.22° and 35.78°, respectively. However, please note that this is an assumption based on the given information.