Revolution: Sum of the angles around a point, equal to Angles that share a vertex and a common side.

Transcription

1 uclidean Geometry Key oncepts lassifying ngles Parallel lines and transversal lines lassifying Triangles Properties of Triangles Relationship between angles ongruency Similarity Pythagoras Mid-point Theorem Properties of Quadrilaterals Terminology cute angle: Greater than 0 0 but less than 90 0 Right angle: ngle equal to 90 0 Obtuse angle: ngle greater than 90 0 Straight angle: ngle equal to Refle angle: ngle greater than but less than Revolution: Sum of the angles around a point, equal to djacent angles: ngles that share a verte and a common side. Vertically opposite angles: ngles opposite each other when two lines intersect. They share a verte and are equal. Supplementary angles: Two angles that add up to omplementary angles: Two angles that add up to Parallel lines Lines that are always the same distance apart transversal line line that intersects two or more parallel lines. Interior angles terior angles orresponding angles ngles that lie in between the parallel lines. ngles that lie outside the parallel lines. ngles on the same side of the lines and the same side of the transversal.

2 o-interior angles ngles that lie in between the lines and on the same side of the transversal. lternate interior angles Interior angles that lie inside the line and on opposite sides of the transversal. X-planation Properties of the angles formed by a transversal line intersecting two parallel lines If the lines are parallel the corresponding angles will be equal the co-interior angles are supplementary the alternate interior angles will be equal. If the corresponding angles will be equal or the co-interior angles are supplementary or the alternate interior angles will be equal lassifying Triangles the lines are parallel There are four kinds of triangles: Scalene Triangle No sides are equal in length Isosceles Triangle Two sides are equal ase angles are equal quilateral Triangle ll three sides are equal ll three interior angles are equal Right-angled triangle One interior angle is

3 Relationship between angles Sum of the angles of a triangle c terior angle of a triangle b a b a a b c 180 c a b c ongruency of triangles Rule 1 Two triangles are congruent if three sides of one triangle are equal in length to the three sides of the other triangle. (SSS) Rule 2 Two triangles are congruent if two sides and the included angle are equal to two sides and the included angle of the other triangle. (SS) Rule 3 Two triangles are congruent if two angles and one side are equal to two angles and one side of the other triangle. (S) Rule 4 Two right-angled triangles are congruent if the hypotenuse and a side of the one triangle is equal to the hypotenuse and a side of the other triangle. (RHS)

4 Similarity Rule 1 () If all three pairs of corresponding angles of two triangles are equal, then the triangles are similar. Rule 2 (SSS) If all three pairs of corresponding sides of two triangles are in proportion, then the triangles are similar. The Theorem of Pythagoras or or Mid-Point Theorem The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the length of the third side. Properties of quadrilaterals Trapezium Two sides are parallel.

5 Parallelogram Opposite sides parallel and equal. Opposite angles equal. iagonals bisect each other. Rectangle Opposite sides parallel and equal in length. iagonals are equal in length and bisect each other. Interior angles are right angles. Rhombus Opposite sides are parallel. ll sides equal in length. iagonals bisect each other at right angles. iagonals bisect the opposite angles. Square Opposite sides parallel. ll sides equal in length. iagonals are equal in length. iagonals bisect each other at right angles. Interior angles are right angles. iagonals bisect interior angles (each bisected angle equals 45 ) Kite djacent pairs of sides are equal in length The longer diagonal bisects the opposite angles. The longer diagonal bisects the other diagonal. The diagonals intersect at right angles.

6 X-ample Questions Question 1 alculate the size of the angles marked with small letters: (a) 49 y (b) 70 (c) (d) Question 2: alculate the size of the angles marked with small letters: (a) 80 (b) 30 y 40

7 Question 3: Prove that Question 4: onsider the diagram below. Is Δ ΔF? Give reasons for your answer. Question 5: In ΔMNP, M 90 0, S is the mid-point of MN and T is the mid-point of NR. (a) Prove U is the mid-point of NP. (b) If ST 4 cm and the area of ΔSNT is 6 cm 2, calculate the area of ΔMNR. (c) Prove that the area of ΔMNR will always be four times the area of ΔSNT, let ST units and SN y units.

8 Question 6 (a) Using the information provided on the diagram, prove that. (b) What type of quadrilateral is? Give a reason Question 7 In the diagram, PQRS is a parallelogram. ˆP 1 and PR bisects ˆR Prove that PQRS is a square. Question 8 Prove that is a trapezium. Question 9 is a parallelogram with diagonal. Given that F H, show that: ΔF Ξ ΔH

9 X-ercise 1. alculate the value of a and b 2. Find the value of 10 24