Find the values of the angles shown in the diagram using rules of geometry

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mathsworkmusic | (Level 2) Educator

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The lines n and m are perpendicular (that is what the symbol between them means). This is to say that they are at right angles to each other.

The angles m1 and m2 are between n and m, and therefore are both right angles, ie 90 ` `degrees` `.

The angles m3 and m4 are adjacent angles on a straight line (line m) and so sum to 180 degrees or a half-turn. Since m3 = 54 degrees, then m4 = 180 - 54 = 126 degrees.

The angles m5 and m6 are opposite respectively to m3 and m4, and so are respectively equal or congruent to m3 and m4, so that m5 = 54 degrees and m6 = 126 degrees.

The angles m7 and m3 are Z angles, and so are equal or congruent, meaning that m7 = 54 degrees also.

Since lines n and l are perpendicular, then the angle between them m8 is a right angle, so that m8 = 90 degrees.

Angles m7 and m11 are complementary, that is, they sum to a right angles, so that m11 = 90 - m7 = 36 degrees.

Angles m9 and m11 are opposite each other, so that m9 = m11 = 36 degrees.

Angles m10 and m9 are adjacent on the line n and so sum to 180 degrees. Since m9 = 36 degrees, this implies that m10 = 180 - m9 = 144 degrees.

m1 = m2 = m8 = 90 degrees

m3 = m5 = m7 = 54 degrees

m4 = m6 = 126  degrees

m9 = m11 = 36 degrees

m10 = 144 degrees

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rachellopez's profile pic

rachellopez | Student, Grade 12 | (Level 1) Valedictorian

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First off, you are told that lines l and n are perpendicular and lines m and n are perpendicular. This means that right angles are created, meaning angle 2 and 8 are 90 degrees. 

You are also told that the measure of angle 3 is 54 degrees. The three angles in a triangle add up to 180, so by default the measure of angle 11 is 36 degrees (90+54+x=180). You can verify that this is correct because angle 11 and 7 form a complementary angle, adding up to 90 degrees (54+36=90).

To solve for angle 1 you can subtract angle 2 from 180 (because angles 1 and 2 form a straight line, or supplementary angle, with a 180 degree measure). This means angle 1 is 90 degrees.  

To find angle 4 you can do the same as with angle 1, except you use angle 3. Angle 4 ends up being 126 degrees. Angle 6 will also be 126 degrees because of the vertical angle rule. This rule also applies to angle 3 and 5, making angle 5 54 degrees also.

Angle 9 is vertical to angle 11 making angle 9 equal to 36 degrees. Lastly, angle 10 can be solved by subtracting 36 from 180 (because 9 and 10 are supplementary), making angle 10 144 degrees. 

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Angle 1 = 90

Angle 2 = 90

Angle 3 = 54

Angle 4 = 126

Angle 5 = 54

Angle 6 = 126

Angle 7 = 54

Angle 8 = 90

Angle 9 = 36

Angle 10 = 144

Angle 11 = 36

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