highschool/Grade 10/Math/MPM2DZ/Unit 4: Trigonometry.md

Angle Theorems

1. Transversal Parallel Line Theorems (TPT)
   1. Alternate Angles are Equal (Z-Pattern)
      
   2. Corresponding Angles Equal (F-Pattern)
      
   3. Interior Angles add up to 180 (C-Pattern)

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2. Supplementary Angle Triangle (SAT)
   

• When two angles add up to 180 degrees

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3. Opposite Angle Theorem (OAT) (OAT)
   

• Two lines intersect, two angles form opposite. They have equal measures

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4. Complementary Angle Theorem (CAT)
   

• The sum of two angles that add up to 90 degrees

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5. Angle Sum of a Triangle Theorem (ASTT)
   

• The sum of the three interior angles of any triangle is 180 degrees

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6. Exterior Angle Theorem (EAT)

• The measure of an exterior angle is equal to the sum of the measures of the opposite interior angles

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7. Isosceles Triangle Theorem (ITT)
   

• The base angles in any isosceles triangle are equal

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8. Sum of The Interior Angle of a Polygon

• The sum of the interioir angles of any polygon is 180(n-2) or 180n - 360, where n is the number of sides of the polygon

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9. Exterior Angles of a Convex Polygon

• The sum of the exterior angle of any convex polygon is always 360 degrees

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Congruency

Congruent: Same size and shape

Side-Side-Side (SSS)

If three sides of a triangle are respectively equal to the three sides of another triangle, then the triangles are congruent

Side-Angle-Side (SAS)

If two sides and the contained angle of a triangle are respectively equal to two sides and the contained angle of another triangle, then the triangles are congruent.

Angle-Side-Angle (ASA)

If two angles and the contained side of a triangle are respectively equal to two angles and the contained side of another triangle, then the triangles are congruent.

Similary Triangles

Similar: Same shape but different sizes (one is an enlargement of the other)

Properties

Lets say we have \(`\triangle ABC \sim \triangle DEF`\) 1. Corresponding angles are equal - \(`\angle A = \angle D`\) - \(`\angle B = \angle E`\) - \(`\angle C = \angle F`\)

2. Corresponding side are proportional.

• \(`\dfrac{AB}{DE} = \dfrac{AC}{DF} = \dfrac{BC}{EF}`\)

3. Proportional Area

• Let \(`k`\) be the scale factor, when concerning for triangle area, if the triangle area can be defined as \(`\dfrac{bh}{2}`\), then by using the smaller triangles side lengths our big triangle’s area is equal to \(`\dfrac{k^2bh}{2}`\). Similar equations and agruments can be dervied from this

Side-Side-Side similarity (RRR \(`\sim`\))

Three pairs of corresponding sides are in the same ratio

Side Angle Side similarity (RAR \(`\sim`\))

Two pairs of corresponding sides are proportional and the contained angle are equal.

Angle-Angle similarity (AA \(`\sim`\))

Two pairs of corresponding angles are equal.