Grade 10 Academic (MPM2D) Unit 2: Analytic Geometry Properties of Triangles and Quadrilaterals Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved

Properties of Triangles Pythagorean Theorem used to find the length of sides of a right triangle Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Types of Triangles Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Some Triangles Theorems Sum of Angles in a Triangle Theorem (SATT) Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Some Triangles Theorems Exterior Angle Theorem (EAT) Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Some Triangles Theorems Isosceles Triangle Theorem (ITT) Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Some Angles Theorems Complementary Angles (CA) Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Some Angles Theorems Supplementary Angles (SA) Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Some Angles Theorems Opposite Angle Theorem(OAT) Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Some Angles Theorems Sum of Angles in a Triangle Theorem(SATT) Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Parallel Lines Theorems Alternate Angles (PLT-Z) Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Parallel Lines Theorems Corresponding Angles (PLT-F) Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Parallel Lines Theorems Interior Angles (PLT-C) Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Properties of Triangles Medians: line from vertex to the midpoint of the opposite side Midpoint: point on a line that divides it into 2 equal parts to draw a median  find mid point of opposite side and connect to vertex Altitude: shows the height of a polygon to draw an altitude  line from vertex to opposite side so that it meets at 90° Perpendicular bisectors: perpendicular to a line segment and meets at its midpoint. to draw a perpendicular bisector  find midpoint, then measure a right angle, draw line.   Angle bisectors: The (interior) bisector of an angle, also called the internal angle bisector is the line or line segment that divides the angle into two equal parts  Centroid: where all 3 medians meet divides each median in the ratio 1: 2 it is the centre of mass Orthocentre: where all 3 altitudes meet Circumcentre: where all 3 perpendicular bisectors meet centre of the circle that passes through the vertices of the triangle the circle is called circumcircle or circumscribed circle Incentre: where all 3 angle bisectors meet centre of the circle that meets each side once the circle is called incircle or inscribed circle Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Median & Centroid Medians: line from vertex to the midpoint of the opposite side Centroid: where all 3 medians meet divides each median in the ratio 1: 2 it is the centre of mass Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Altitude & Orthocentre shows the height of a polygon to draw an altitude  line from vertex to opposite side so that it meets at 90° Orthocentre: where all 3 altitudes meet Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Perpendicular Bisector & Circumcentre Perpendicular bisectors: perpendicular to a line segment and meets at its midpoint. to draw a perpendicular bisector  find midpoint, then measure a right angle, draw line.   Circumcentre: where all 3 perpendicular bisectors meet centre of the circle that passes through the vertices of the triangle the circle is called circumcircle or circumscribed circle Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Angle Bisector & Incentre Angle bisectors: The (interior) bisector of an angle, also called the internal angle bisector is the line or line segment that divides the angle into two equal parts  Incentre: where all 3 angle bisectors meet centre of the circle that meets each side once the circle is called incircle or inscribed circle Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Properties of Quadrilaterals What is a quadrilateral? Polygon with 4 sides and sum of the angles is 360o Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Properties of Quadrilaterals Picture Description & Properties Square Quadrilateral with 4 sides are equal All angles are 90o Diagonals are perpendicular to each other Rectangle   Quadrilateral with 2 pairs of equal opposite sides Diagonals are not perpendicular to each other Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Properties of Quadrilaterals Picture Description & Properties Rhombus   Quadrilateral with 4 sides are equal None of angles is 90o A pair of acute angles and a pair of obtuse angles Diagonals are perpendicular to each other Parallelogram Quadrilateral with 2 pairs of equal opposite sides Diagonals are not perpendicular to each other Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Properties of Quadrilaterals Picture Description & Properties Kite Quadrilateral with 2 pairs of equal adjacent sides Diagonals are perpendicular to each other Trapezoid Quadrilateral with 1 pair of unequal but parallel opposite sides   Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

Homework Work sheet: Cartesian Triangles - A Summary Text: Read P. 33 & 38. Make Summary notes P. 34    #1 - 4 P. 39    #1, 2, 4, 5, 6 Read P. 36. Make Summary notes P. 37    #1, 2, 3, 4, 6, 7 Check the website for updates Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved

End of lesson Properties of Triangles and Quadrilaterals © 2017 E. Choi – MPM2D - All Rights Reserved