Question

find the indicated angle measure.

1. find m∠lmn
2. ∠ghk is a straight angle. find m∠jhk
3. m∠abc = 95°. find m∠abd and m∠dbc
4. m∠xyz = 117°. find m∠xyw and m∠wyz
5. ∠lmn is a straight angle. find m∠lmp and m∠nmp
6. ∠abc is a straight angle. find m∠abx and m∠cbx
7. find m∠rsq and m∠tsq
8. find m∠deh and m∠feh

Explanation:

Step1: Identify angle relationships

Determine if angles are part of a straight line, right - angle, or other known geometric relationships. For example, if an angle is a straight angle, its measure is \(180^{\circ}\).

Step2: Set up equations

Based on the angle relationships, set up equations. If an angle \(\angle A\) is divided into \(\angle B\) and \(\angle C\) by a ray, then \(m\angle A=m\angle B + m\angle C\).

Step3: Solve equations

If the equations involve variables, solve for the variables using algebraic operations such as addition, subtraction, multiplication, and division. Then substitute the value of the variable back into the expressions for the angles to find their measures.

Answer:

Since the problem is incomplete (no specific values or relationships given for most of the angle - finding tasks other than some general statements about angles like straight - angle), we cannot provide specific numerical answers. However, for a general approach to these types of problems:

1. If \(\angle GHK\) is a straight angle (\(180^{\circ}\)), and we want to find \(m\angle JHK\), we need more information about the relationship between \(\angle GHJ\) and \(\angle JHK\).
2. Given \(m\angle ABC = 95^{\circ}\) and we want to find \(m\angle ABD\) and \(m\angle DBC\), if \(BD\) is a ray within \(\angle ABC\), and say \(m\angle ABD=x\) and \(m\angle DBC = y\), then \(x + y=95^{\circ}\). If additional information like a ratio between \(x\) and \(y\) is given, we can solve for \(x\) and \(y\).
3. If \(\angle LMN\) is a straight angle (\(180^{\circ}\)) and we want to find \(m\angle LMP\) and \(m\angle NMP\), if \(m\angle LMP = a\) and \(m\angle NMP=b\), then \(a + b = 180^{\circ}\). If we know, for example, that \(a=3x + 10\) and \(b = 2x+20\), then \((3x + 10)+(2x + 20)=180\).

• First, simplify the left - hand side: \(3x+10 + 2x+20=5x + 30\).
• Then, set up the equation \(5x+30 = 180\).
• Subtract 30 from both sides: \(5x=180 - 30=150\).
• Divide both sides by 5: \(x = 30\).
• So \(m\angle LMP=3x + 10=3\times30+10 = 100^{\circ}\) and \(m\angle NMP=2x + 20=2\times30+20 = 80^{\circ}\).

In general, for problems involving finding angle measures:

1. Use the properties of angles such as the sum of angles in a straight line (\(180^{\circ}\)), sum of angles in a right - angle (\(90^{\circ}\)), and angle - addition postulate (if an angle is divided into two non - overlapping sub - angles, the measure of the whole angle is the sum of the measures of the sub - angles).