Question

1. given lines p and q intersect at one point, complete the sentence below then solve for all values x and y algebraically. *if two angles are a linear - pair, then
2. given rays l, q, m, and n intersect at a common point, complete the sentence below then solve for x. *the sum of adjacent angles that meet at any one vertex is
3. find the measure of each angle in the diagrams below: *if a point is in the interior of an angle, then the big angle is equal to the ______ of the two smaller angles a) find m∠wvu if m∠zvu = 62° and m∠wvz = 50°. b) m∠zhg = 11x - 1, m∠ihz = 24°, and m∠ihg = 12x + 13. find m∠ihg.

Explanation:

Step1: Recall linear - pair property

If two angles are a linear pair, then they are supplementary (their sum is 180°). For the first part with \(y\) - related angles \(y^{2}+20y\) and \(- 2y + 36\), we set up the equation \(y^{2}+20y-2y + 36=180\). Rearrange to get \(y^{2}+18y - 144 = 0\). Factor: \((y + 24)(y - 6)=0\). So \(y=-24\) or \(y = 6\).

Step2: Recall angle - sum property at a vertex

The sum of adjacent angles that meet at any one vertex is 360°. So for the angles \(\frac{1}{2}x + 23\), \(6x+9\), \(5x + 13\) and 131, we set up the equation \((\frac{1}{2}x + 23)+(6x+9)+(5x + 13)+131 = 360\). Combine like - terms: \(\frac{1}{2}x+6x + 5x+23 + 9+13+131 = 360\), \(\frac{1 + 12+10}{2}x+176 = 360\), \(\frac{23}{2}x=360 - 176\), \(\frac{23}{2}x = 184\), \(x = 16\).

Step3: Recall angle - addition postulate

If a point is in the interior of an angle, then the big angle is equal to the sum of the two smaller angles.

a)

\(m\angle WVU=m\angle WVZ + m\angle ZVU\). Substitute the given values: \(m\angle WVU=50^{\circ}+62^{\circ}=112^{\circ}\).

b)

Since \(m\angle IHG=m\angle IHZ + m\angle ZHG\), we set up the equation \(12x + 13=24+(11x - 1)\). Simplify: \(12x+13=11x + 23\), \(12x-11x=23 - 13\), \(x = 10\). Then \(m\angle IHG=12x + 13=12\times10+13 = 133^{\circ}\).

Answer:

1. If two angles are a linear pair, then they are supplementary; \(y=-24\) or \(y = 6\)
2. The sum of adjacent angles that meet at any one vertex is 360°; \(x = 16\)
3. a) 112°

b) 133°