What lets you say two angles are equal when a line crosses parallel lines?

Alternate angles are equal, and so are corresponding angles . Naming which of the two you are using is the reason that has to accompany the fact.

In a proof, what does knowing two sides of a triangle are equal give you?

The triangle is isosceles , so its two base angles are equal as well. Marking that on the diagram early often makes the rest of the proof visible.

True or False? Similar shapes have equal angles and equal side lengths.

False. Similar shapes have equal angles, but their sides are in the same ratio rather than equal. Shapes with equal angles and equal sides are congruent , meaning identical.

What does the Alternate Segment Theorem say?

The angle between a tangent and a chord equals the angle in the alternate segment . It is the circle theorem students most often fail to spot, because it involves a tangent rather than only chords.

GCSEFurther MathsAQAFlashcards6. GeometryGeometric Proof

Geometric Proof (AQA GCSE Further Maths): Flashcards

Exam code: 8365

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FrontGeometric Proof
What does each step of a geometric proof need?

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Cards in this collection (9)

What does each step of a geometric proof need?
A fact and a reason, so that every angle you write down is justified by a named rule.
Even the most basic facts need their reason given, such as "angles on a straight line add up to $180 °$ ".
How can the angle at $B$ in triangle $A B C$ be written?
The angle can be written as "angle $A B C$ ", always with the vertex letter in the middle.
Vertices and lengths take capital letters, so $A B$ is a side and $A B C$ is either a triangle or an angle.
What lets you say two angles are equal when a line crosses parallel lines?
Alternate angles are equal, and so are corresponding angles.
Naming which of the two you are using is the reason that has to accompany the fact.
In a proof, what does knowing two sides of a triangle are equal give you?
The triangle is isosceles, so its two base angles are equal as well.
Marking that on the diagram early often makes the rest of the proof visible.
True or False?
Similar shapes have equal angles and equal side lengths.
False.
Similar shapes have equal angles, but their sides are in the same ratio rather than equal.
Shapes with equal angles and equal sides are congruent, meaning identical.
Complete these circle theorems:
The angle at the centre is $______$ the angle at the circumference.
Opposite angles in a cyclic quadrilateral add up to $______$ .
The angle in a semicircle is $______$ .
The completed theorems are:
The angle at the centre is twice the angle at the circumference.
Opposite angles in a cyclic quadrilateral add up to $180 °$ .
The angle in a semicircle is a right angle.
Angles in the same segment are also equal, a triangle formed by two radii is isosceles, and two tangents from the same point are equal in length.
What does the Alternate Segment Theorem say?
The angle between a tangent and a chord equals the angle in the alternate segment.
It is the circle theorem students most often fail to spot, because it involves a tangent rather than only chords.
Why might a circle-theorem proof not require you to find $x$ and $y$ ?
Because the question asks only for a relationship, such as showing that $x + y = 90$ .
Finding the individual values may be impossible, so keep the target result in mind and aim straight at it.
$A B C D$ is a cyclic quadrilateral with a tangent touching the circle at $D$ . Angle $A B C$ is $w$ , angle $C D Q$ is $x$ and angle $A C D$ is $y$ . Prove $w = x + y$ .
Opposite angles in a cyclic quadrilateral give angle $A D C = 180 - w$ , and the Alternate Segment Theorem gives angle $D A C = x$ .
The angles of triangle $A D C$ sum to $180 °$ , so $(180 - w) + x + y = 180$ , which rearranges to $w = x + y$ .

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