Living and Learning: Maths Geometry - Alternate and Corresponding angles (KS3)

Here we looked at parallel lines that are bisected by a transverse line (sounds complicated he he)

We look at Z-angles which are alternate and =
We look at F-angles which are corresponding and =

congruent means =

Opposite angles of a parallelogram are congruent. (=)

Remember the definition of parallelogram: a quadrilateral that thas two pairs of opposite parallel sides.

Triangles can be used to prove this rule about the opposite angle.

Consecutive angles are the ones that are adjacent (next to each other)

Consecutive angles are supplementary.

Then we have to use this to prove that angles in triangles and parallelograms are = using these rules.

The proving works by drawing and labelling the shape and then adding additional lines which are parallel (Triangle) or a diagonal (parallelogram) and using these rules to prove that internal angles of a triangle add up to 180 degree or that opposite angles in a parallelogram are =.

There are ways to write your working out in geometry and how to represent side AB or angle ABC or refer to triangle ABC.

You need to write your proof in a certain way and order to prove what you are required to prove.

Triangle Proof:
Prove that the angles in a triangle add up to 180 degrees

1. Draw the triangle
2. Let the three angles be denoted be letters a,b,c
3. Now draw a line through the top vertex (Corner) parallel to the baseline.
4. Add in angles d & e on either side of angle a.
5. a + b + e = 180 degrees (angles on a straight line = 180 degrees)
6. but d = b (alternate angles)
7. therefore a + b + c = 180 degrees.

Parallellogram - = Angles Proof

Here are the details from another source: Prove that the opposite angles of a parallelogram are equal.

1.Firstly look at the actual Theorem

- The opposite sides and angles of a parallelogram are = to one another.

- Both of its diameters bisect its area.

2. Construct a parallelogram ACDB or ACDB is a parallelogram

3. Construct BC to be a segment

4. Line AB is parallel to Line CD

Reason: Definition of a parallegoram.

5. Parallel means that alternate angles are =

6. So ∠ABC=∠BCD

7. AC∥BD. Parallelogram definition.

8. Parallel means that alternate angles are =

9. So ∠ACB=∠CBD

10. Line CB = Line BC (shared line)
11 ASA Proof (2 triangles both have 2 equal /congruent angles and an included side. In this case they share the common side CB.

12. So the remaining opposite angles will be equal to each other. (ASA)

This proves that the opposite angles and side of a parallelogram are equal/congruent.

This is based on the ASA Postulate see: http://www.mathwarehouse.com/geometry/congruent_triangles/angle-side-angle-postulate.php

Sources: http://www.mathwarehouse.com/geometry/quadrilaterals/parallelograms/

http://www.proofwiki.org/wiki/Opposite_Sides_and_Angles_of_Parallelogram_are_Equal

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We also have a book called Help your kids with Maths by Carol Vorderman, which we got out the library.