Question

angle diagrams labeled a to j (each with angle expressions: a: (17x+21)°, 123°; b: (14x-25)°, 51°; c: (10x-26)°, (7x+18)°; d: (2x+70)°, (101-5x)°; e: 27°, (8x-1)°; f: (7x-44)°, 75°, 63°; g: 107°, 85°, (4-2x)°; h: (11x-23)°, 41°; i: (5x-3)°, 17°, (13x+4)°; j: (3y+10)°, (8x+1)°, 62°, (5x-24)°) and a blue grid with labels g, b, d, i.

Answer:

To solve these angle - related problems, we will use the properties of angles such as supplementary angles (sum to \(180^{\circ}\)), vertical angles (equal), and complementary angles (sum to \(90^{\circ}\)) depending on the diagram. Let's take each part one by one:

Part A

We know that the angle of \(123^{\circ}\) and \((17x + 21)^{\circ}\) are supplementary (they form a linear pair, so their sum is \(180^{\circ}\)).

Step 1: Set up the equation

\(123+(17x + 21)=180\)

Step 2: Simplify the left - hand side

\(17x+144 = 180\)

Step 3: Subtract 144 from both sides

\(17x=180 - 144\)
\(17x = 36\)

Step 4: Solve for x

\(x=\frac{36}{17}\approx2.12\)

Part B

The angle \((14x - 25)^{\circ}\) and \(51^{\circ}\) are supplementary (linear pair).

Step 1: Set up the equation

\((14x - 25)+51 = 180\)

Step 2: Simplify the left - hand side

\(14x + 26=180\)

Step 3: Subtract 26 from both sides

\(14x=180 - 26\)
\(14x = 154\)

Step 4: Solve for x

\(x = 11\)

Part C

The two angles \((10x - 26)^{\circ}\) and \((7x + 18)^{\circ}\) are vertical angles, so they are equal.

Step 1: Set up the equation

\(10x-26=7x + 18\)

Step 2: Subtract \(7x\) from both sides

\(10x-7x-26=7x - 7x+18\)
\(3x-26 = 18\)

Step 3: Add 26 to both sides

\(3x=18 + 26\)
\(3x = 44\)

Step 4: Solve for x

\(x=\frac{44}{3}\approx14.67\)

Part D

The two angles \((2x + 70)^{\circ}\) and \((101 - 5x)^{\circ}\) are vertical angles, so they are equal.

Step 1: Set up the equation

\(2x + 70=101-5x\)

Step 2: Add \(5x\) to both sides

\(2x+5x + 70=101-5x + 5x\)
\(7x+70 = 101\)

Step 3: Subtract 70 from both sides

\(7x=101 - 70\)
\(7x = 31\)

Step 4: Solve for x

\(x=\frac{31}{7}\approx4.43\)

Part E

The angle of \(27^{\circ}\), \((8x - 1)^{\circ}\) and the right angle (if we assume the straight line forms a right angle? Wait, no, the sum of the three angles should be \(180^{\circ}\) (linear pair). Wait, the diagram shows three angles: \(27^{\circ}\), \((8x - 1)^{\circ}\) and a right angle? Wait, no, looking at the diagram, the three angles are adjacent and form a straight line. So \(27+(8x - 1)+90 = 180\) (assuming one angle is a right angle? Wait, maybe the sum of the three angles is \(180^{\circ}\)).

Step 1: Set up the equation

\(27+(8x - 1)+90=180\)

Step 2: Simplify the left - hand side

\(8x+116 = 180\)

Step 3: Subtract 116 from both sides

\(8x=180 - 116\)
\(8x = 64\)

Step 4: Solve for x

\(x = 8\)

Part F

The sum of the angles around a point? Wait, the angles \(75^{\circ}\), \(63^{\circ}\) and \((7x - 44)^{\circ}\) and the right angle? Wait, the sum of angles around a point is \(360^{\circ}\), but if we consider the angles on a straight line, the sum of adjacent angles on a straight line is \(180^{\circ}\). Wait, looking at the diagram, the angles \(75^{\circ}\), \(63^{\circ}\) and \((7x - 44)^{\circ}\) and the vertical angle? Wait, maybe the sum of \(75 + 63+(7x - 44)=180\) (since they are on a straight line).

Step 1: Set up the equation

\(75+63+(7x - 44)=180\)

Step 2: Simplify the left - hand side

\(7x + 94=180\)

Step 3: Subtract 94 from both sides

\(7x=180 - 94\)
\(7x = 86\)

Step 4: Solve for x

\(x=\frac{86}{7}\approx12.29\)

Part G

The sum of the angles \(107^{\circ}\), \(85^{\circ}\) and \((4 - 2x)^{\circ}\) and the vertical angle? Wait, the sum of angles around a point is \(360^{\circ}\), but if we consider the angles on a straight line, the sum of adjacent angles on a straight line is \(180^{\circ}\). Wait, \(107+85+(4 - 2x)=180\)

Step 1: Set up the equation

\(196-2x = 180\)

Step 2: Subtract 196 from both sides

\(- 2x=180 - 196\)
\(-2x=-16\)

Step 3: Solve for x

\(x = 8\)

Part H

The sum of the angles \((11x - 23)^{\circ}\), \(41^{\circ}\) and the vertical angles? Wait, the sum of angles around a point is \(360^{\circ}\), and vertical angles are equal. Let's assume that the angle \((11x - 23)^{\circ}\) and its vertical angle are equal, and the angle \(41^{\circ}\) and its vertical angle are equal. So \(2(11x - 23)+2\times41=360\)

Step 1: Simplify the equation

\(22x-46 + 82=360\)
\(22x + 36=360\)

Step 2: Subtract 36 from both sides

\(22x=360 - 36\)
\(22x = 324\)

Step 3: Solve for x

\(x=\frac{324}{22}=\frac{162}{11}\approx14.73\)

Part I

The two angles \((5x - 3)^{\circ}\) and \((13x + 4)^{\circ}\) and \(17^{\circ}\). Wait, the sum of angles around a point is \(360^{\circ}\), and vertical angles are equal. Let's assume that the angle \((5x - 3)^{\circ}\) and its vertical angle are equal, and the angle \((13x + 4)^{\circ}\) and its vertical angle are equal, and the angle \(17^{\circ}\) and its vertical angle are equal. So \(2(5x - 3)+2(13x + 4)+2\times17=360\)

Step 1: Simplify the equation

\(10x-6 + 26x+8 + 34=360\)
\(36x + 36=360\)

Step 2: Subtract 36 from both sides

\(36x=360 - 36\)
\(36x = 324\)

Step 3: Solve for x

\(x = 9\)

Part J

The sum of the angles \((3y + 10)^{\circ}\), \((8x + 1)^{\circ}\), \(62^{\circ}\) and \((5x - 24)^{\circ}\) around a point is \(360^{\circ}\). Wait, maybe we have vertical angles. Let's assume that \((3y + 10)^{\circ}\) and \((5x - 24)^{\circ}\) are vertical angles, and \((8x + 1)^{\circ}\) and \(62^{\circ}\) are vertical angles.

First, for the angles \((8x + 1)^{\circ}\) and \(62^{\circ}\) (vertical angles, so equal):

Step 1: Set up the equation for x

\(8x+1 = 62\)

Step 2: Subtract 1 from both sides

\(8x=62 - 1\)
\(8x = 61\)

Step 3: Solve for x

\(x=\frac{61}{8}=7.625\)

For the angles \((3y + 10)^{\circ}\) and \((5x - 24)^{\circ}\) (vertical angles, so equal). Substitute \(x = \frac{61}{8}\) into \(5x-24\):

\(5\times\frac{61}{8}-24=\frac{305}{8}-24=\frac{305 - 192}{8}=\frac{113}{8}\)

Then set up the equation for y:

Step 1: Set up the equation for y

\(3y + 10=\frac{113}{8}\)

Step 2: Subtract 10 from both sides

\(3y=\frac{113}{8}-10=\frac{113 - 80}{8}=\frac{33}{8}\)

Step 3: Solve for y

\(y=\frac{11}{8}=1.375\)

Since you haven't specified which part (A - J) you want to solve, here are the solutions for each:

• A: \(x=\frac{36}{17}\approx2.12\)
• B: \(x = 11\)
• C: \(x=\frac{44}{3}\approx14.67\)
• D: \(x=\frac{31}{7}\approx4.43\)
• E: \(x = 8\)
• F: \(x=\frac{86}{7}\approx12.29\)
• G: \(x = 8\)
• H: \(x=\frac{162}{11}\approx14.73\)
• I: \(x = 9\)
• J: \(x=\frac{61}{8}=7.625\), \(y=\frac{11}{8}=1.375\)