6.3 Geometry: Angles, Triangles, Area, Perimeter, and Circles

Why This Matters for the PUPCET

Geometry questions in the PUPCET Mathematics portion are formula-driven and highly predictable. Unlike word problems, they rarely hide the setup: you are shown a figure or given dimensions, and the challenge is recalling the correct relationship and executing it cleanly. Students who memorize a small set of angle rules, the Pythagorean triples, and one formula table can answer most geometry items in under a minute — a huge advantage when the whole exam allows barely more than that per question.

Angle Relationships

An angle measures the opening between two rays that share an endpoint, in degrees (°). Four relationships cover nearly every PUPCET angle item.

Parallel Lines Cut by a Transversal

When a transversal (a line crossing two parallel lines) slices through, eight angles form, and they come in equal pairs:

• Corresponding angles (same position at each crossing) are equal.
• Alternate interior angles (inside the parallel lines, opposite sides of the transversal) are equal.
• Alternate exterior angles (outside, opposite sides) are equal.
• Consecutive (same-side) interior angles are supplementary — they sum to 180°.

The practical rule: when lines are parallel, every acute angle in the figure equals every other acute angle, every obtuse angle equals every other obtuse angle, and any acute plus any obtuse equals 180°. If one angle is 72°, then four of the eight angles are 72° and the other four are 108°.

Worked Example 1: Parallel Lines

Two parallel lines are cut by a transversal. One interior angle measures 118°. Find its same-side interior partner and its alternate interior partner.

The same-side (consecutive) interior angle is supplementary: 180° − 118° = 62°. The alternate interior angle is equal: 118°.

Triangles

A triangle is a three-sided polygon. Its single most important property:

The three interior angles of every triangle sum to exactly 180°.

Two more rules the PUPCET uses regularly: an exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles, and the triangle inequality says any two sides must sum to MORE than the third side — so sides of 3, 4, and 9 cannot form a triangle because 3 + 4 < 9.

Worked Example 2: Missing Angle

In triangle ABC, angle A = 47° and angle B = 83°. Find angle C.

C = 180° − 47° − 83° = 50°.

Worked Example 3: Isosceles Triangle

An isosceles triangle has a vertex angle of 40°. Find each base angle.

The base angles are equal, so each is (180° − 40°) ÷ 2 = 140° ÷ 2 = 70°.

The Pythagorean Theorem

In a right triangle, the two legs a and b and the hypotenuse c (the side opposite the right angle, always the longest) satisfy:

a² + b² = c²

This works only for right triangles. Instead of computing squares every time, memorize the Pythagorean triples — whole-number side sets that appear constantly on exams:

Worked Example 4: Find the Hypotenuse

A right triangle has legs of 9 cm and 12 cm. Find the hypotenuse.

Slow way: c² = 81 + 144 = 225, so c = 15 cm. Fast way: 9-12-? is 3 × (3-4-5), so the hypotenuse is 3 × 5 = 15 cm instantly.

Worked Example 5: Find a Missing Leg

The hypotenuse is 13 m and one leg is 5 m. Find the other leg.

Recognize the 5-12-13 triple: the missing leg is 12 m. Verify: 25 + 144 = 169 = 13². ✓

Perimeter, Area, and Circle Formulas

Perimeter is the total distance around a figure; area is the surface it covers, always in square units. This table is the heart of PUPCET geometry — memorize it completely.

Use π ≈ 3.1416, or 3.14, or 22/7 when the radius is a multiple of 7 — PUPCET options usually reveal which approximation they used. Watch the classic traps: area of a triangle needs the PERPENDICULAR height, not a slanted side; circle area uses the radius, so if you are given the diameter, halve it first.

Worked Example 6: Rectangle

A school garden in Quezon City is 14 m long and 9 m wide. Find its perimeter and area.

Perimeter: 2(14) + 2(9) = 28 + 18 = 46 m. Area: 14 × 9 = 126 m².

Worked Example 7: Trapezoid

A trapezoidal lot has parallel sides of 10 m and 16 m, with a height of 6 m. Find its area.

A = ½(10 + 16)(6) = ½(26)(6) = 13 × 6 = 78 m². The trap is adding the bases and multiplying by height without the ½, which doubles the answer to 156 — and that value will be among the choices.

Worked Example 8: Circle Given the Diameter

A round table has a diameter of 14 dm. Find its circumference and area (use π = 22/7).

Radius first: r = 14 ÷ 2 = 7 dm. Circumference: C = πd = (22/7)(14) = 44 dm. Area: A = πr² = (22/7)(7²) = (22/7)(49) = 22 × 7 = 154 dm². If you mistakenly plug the diameter into πr², you get 616 dm² — four times too large, because area grows with the square of the radius.

Multi-Step Problem: Putting It Together

A rectangular plaza measures 20 m by 14 m, with a circular fountain of radius 3 m in the center. What area of the plaza remains for walking? (Use π = 3.14.)

Step 1. Rectangle area: 20 × 14 = 280 m². Step 2. Fountain area: πr² = 3.14 × 9 = 28.26 m². Step 3. Subtract: 280 − 28.26 = 251.74 m².

This "shaded region" pattern — big area minus small area — is one of the most reliable multi-step items on Philippine entrance exams. Always compute each region separately, then subtract.

Common Traps and Exam Tips

• Using the diameter in πr² instead of halving it first.
• Applying the Pythagorean theorem to a triangle with no right angle.
• Forgetting the ½ in triangle and trapezoid areas.
• Assuming figures are drawn to scale — trust the numbers, never the picture.
• Mixing units: if length is in meters and width in centimeters, convert before multiplying.

Section Takeaway

Angles: complementary 90°, supplementary 180°, vertical angles equal, parallel lines make equal acute angles and equal obtuse angles. Triangles: interior angles sum to 180°, and right triangles obey a² + b² = c² — memorize the 3-4-5 and 5-12-13 families. Formulas: keep the perimeter-area table and the circle pair C = 2πr, A = πr² at your fingertips, and always halve a given diameter before computing.