Math Problem Statement

If m∠2 = 41°, m∠5 = 94°, and m∠10 = 109°, find each measure.

Solution

This problem involves finding the measures of various angles in a geometric figure, using the given values of certain angles and the relationships between them. Here's a step-by-step breakdown:

Given:

  • m2=41m\angle 2 = 41^\circ
  • m5=94m\angle 5 = 94^\circ
  • m10=109m\angle 10 = 109^\circ

We are asked to find the following angle measures:

  • m1m\angle 1
  • m3m\angle 3
  • m4m\angle 4
  • m6m\angle 6
  • m7m\angle 7
  • m8m\angle 8
  • m9m\angle 9

Approach:

a. m1m\angle 1:

Angle 1\angle 1 and 2\angle 2 are adjacent to form a straight line, making them supplementary. Thus: m1+m2=180m\angle 1 + m\angle 2 = 180^\circ m1=18041=139m\angle 1 = 180^\circ - 41^\circ = 139^\circ

b. m3m\angle 3:

Angle 3\angle 3 is vertically opposite to 2\angle 2, so they are equal: m3=m2=41m\angle 3 = m\angle 2 = 41^\circ

c. m4m\angle 4:

Angles 3\angle 3, 4\angle 4, and 5\angle 5 form a triangle. The sum of the angles in a triangle is 180180^\circ: m3+m4+m5=180m\angle 3 + m\angle 4 + m\angle 5 = 180^\circ Substitute the known values: 41+m4+94=18041^\circ + m\angle 4 + 94^\circ = 180^\circ m4=180135=45m\angle 4 = 180^\circ - 135^\circ = 45^\circ

d. m6m\angle 6:

Angle 6\angle 6 and 5\angle 5 form a linear pair, meaning they are supplementary: m6+m5=180m\angle 6 + m\angle 5 = 180^\circ m6=18094=86m\angle 6 = 180^\circ - 94^\circ = 86^\circ

e. m7m\angle 7:

Angle 7\angle 7 is vertically opposite to 6\angle 6, so: m7=m6=86m\angle 7 = m\angle 6 = 86^\circ

f. m8m\angle 8:

Angles 8\angle 8, 9\angle 9, and 10\angle 10 form a triangle, so their sum must be 180180^\circ: m8+m9+m10=180m\angle 8 + m\angle 9 + m\angle 10 = 180^\circ But we don't know m9m\angle 9 yet, so we'll need to solve for it later.

g. m9m\angle 9:

Angle 9\angle 9 and 8\angle 8 form a linear pair, meaning they are supplementary: m9+m8=180m\angle 9 + m\angle 8 = 180^\circ At this stage, we will need more information about either 8\angle 8 or 9\angle 9 to calculate.

Let me know if you would like further assistance or have more specific questions on this. Here are some related problems:

  1. How do you find the third angle in a triangle when two angles are known?
  2. What is the relationship between vertically opposite angles?
  3. How do supplementary angles relate to each other?
  4. How can you use triangle angle sums to solve for unknown angles?
  5. How do you find angle measures for a quadrilateral?

Tip: In geometry problems involving lines and angles, always look for linear pairs and vertical angles to simplify your calculations.

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Math Problem Analysis

Mathematical Concepts

Geometry
Angles
Linear Pairs
Triangles

Formulas

Sum of angles on a straight line = 180°
Sum of angles in a triangle = 180°

Theorems

Supplementary angles
Vertical angles

Suitable Grade Level

Grades 6-8