The Elements of Geometry

2

Step 1: When two circles intersect, they can have at most 2 common points. Step 2: Given that the circles intersect at points R and S, these are the two intersection points. Step 3: Two distinct circles cannot have more than 2 points in common (unless they are identical circles). Step 4: If they had 3 or more points in common, they would be the same circle, which contradicts having different centers P and Q. Step 5: Therefore, the circles have exactly 2 common points: R and S.

AB = EF

Step 1: We are given that AB = CD and EF = CD. Step 2: This situation directly applies Euclid's first axiom: 'things which are equal to the same thing are equal to one another'. Step 3: Since both AB and EF are equal to CD, they must be equal to each other. Step 4: By the transitive property of equality: AB = CD and EF = CD, therefore AB = EF. Step 5: This axiom is fundamental in establishing equality relationships in geometric proofs.

90°

Step 1: Euclid's fourth postulate states that all right angles are equal to one another. Step 2: By modern convention, a right angle measures 90 degrees. Step 3: This postulate establishes that whenever we have a right angle, it always has the same measure regardless of its position or orientation. Step 4: The 90-degree measure comes from dividing a straight line (180°) into two equal parts. Step 5: This standardization allows us to use right angles as reference angles for measuring other angles.

7 cm

Step 1: Given that M is the midpoint of XY, this means M divides XY into two equal parts. Step 2: If XY = 14 cm, then XM = MY (since M is the midpoint). Step 3: Using Euclid's axiom about wholes and parts: XY = XM + MY. Step 4: Since XM = MY, we can write: 14 = XM + XM = 2 × XM. Step 5: Solving for XM: XM = 14 ÷ 2 = 7 cm. Verification: XM + MY = 7 + 7 = 14 cm ✓