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Friday, July 31, 2026

Lines, Angles, and Basic Triangles UPCAT Reviewer

TEACHER ABI UPCAT MATHEMATICS

Lines, Angles, and Basic Triangles

Name the relationship first; calculate only after you know whether angles are equal or supplementary.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Lines, Angles, and Basic Triangles

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Most angle problems reduce to equality or a known total

Vertical angles are equal. A linear pair totals 180°. Corresponding and alternate interior angles are equal when lines are parallel. Triangle interior angles total 180°, and an exterior angle equals the two remote interior angles.

a² + b² = c² for a right triangle

UPCAT geometry diagrams showing vertical angles, triangle angle sum, exterior angles, triangle inequality, and a right triangle — Use the diagram to identify the relationship before writing an equation.

Mark the relationship

Do not assume angles are equal just because a diagram looks symmetric.

Match sides and opposite angles

The largest angle lies opposite the longest side.

DO IT FAST

Label equal angles; circle every 180° total

If vertical angles are 3x + 5 and 5x−19, set them equal. If same-side interior angles are 2x + 10 and 4x + 20, set their sum to 180.

Why it works

The diagram determines the equation; the algebra merely finishes it.

WORKED EXAMPLES

Five forms you should recognize

1. Vertical angles

Problem: Two vertical angles measure 3x + 5 and 5x−19 degrees. Find x.

3x + 5 = 5x−19 → x = 12

Why: Opposite angles formed by intersecting lines are equal.

2. Triangle sum

Problem: Two angles of a triangle measure 47° and 63°. Find the third angle.

47°+63°+x = 180° → x = 70°

Why: A triangle’s interior angles total 180°.

3. Exterior angle

Problem: The remote interior angles of a triangle are 35° and 72°. Find the exterior angle.

35°+72° = 107°

Why: An exterior angle equals the sum of the two remote interior angles.

4. Triangle inequality

Problem: Can segments of lengths 5, 7, and 9 form a triangle?

5 + 7 > 9, 5 + 9 > 7, 7 + 9 > 5

Why: Every pair has a sum greater than the remaining side.

5. Right triangle

Problem: A right triangle has legs 6 and 8. Find its hypotenuse.

c² = 6² + 8² = 100 → c = 10

Why: The Pythagorean theorem applies because the known sides are perpendicular legs.

COMMON TRAPS

Check before you commit

Trusting the drawing instead of stated markings
Confusing vertical with adjacent angles
Forgetting same-side interior angles are supplementary
Using AAA as congruence
Ignoring the triangle inequality
Using the Pythagorean theorem on a non-right triangle

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Lines, Angles, and Basic Triangles FAQ

Are vertical angles adjacent?

No. They are opposite angles formed by intersecting lines.

Does AAA prove congruence?

No. It proves similarity because the triangles may have different sizes.

When can I use a²+b²=c²?

Only when the triangle is right and c is the hypotenuse.

RELATED COMPETENCIES

Continue your mathematics review.

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Mean, Median, Mode, and Weighted Mean UPCAT Reviewer

TEACHER ABI UPCAT MATHEMATICS

Mean, Median, Mode, and Weighted Mean

Choose the summary that matches the data—and keep totals and weights visible.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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The measures answer different questions

Mean is total divided by count. Median is the middle after ordering. Mode is most frequent. A weighted mean multiplies each value by its share before adding.

weighted mean = Σ(value×weight)/Σweights

Mean follows the total

If the mean and number of observations are known, total = mean×count.

Median resists outliers

An extreme value can pull the mean while leaving the middle position nearly unchanged.

DO IT FAST

Write total = mean × count

For a missing score, find the target total first. Mean 18 for 4 values means total 72. If three values total 49, the missing value is 23.

Why it works

Mean problems become ordinary total problems once you recover the sum.

WORKED EXAMPLES

Five forms you should recognize

1. Mean

Problem: A player scored 6, 8, 9, 12, and 15 points in five games. What was the mean score?

(6 + 8 + 9 + 12 + 15)5 = 10

Why: The mean distributes the total of 50 equally across five games.

2. Median

Problem: Find the median of 11, 4, 9, 7, and 15.

4, 7, 9, 11, 15 → median = 9

Why: Median means the middle position after ordering the values.

3. Mode

Problem: What is the mode of 3, 5, 5, 6, 8, 8, 8, and 9?

mode = 8

Why: Eight occurs three times, more frequently than any other value.

4. Weighted grade

Problem: Quizzes count 40% and the exam 60%. A student earned 85 in quizzes and 90 on the exam. What is the final grade?

0.40(85) + 0.60(90) = 88

Why: Multiply each score by its contribution before adding.

5. Combined mean

Problem: Ten students have mean 80 and fifteen students have mean 90. What is their combined mean?

[10(80) + 15(90)]25 = 86

Why: Recover both group totals first; directly averaging 80 and 90 would ignore the different group sizes.

COMMON TRAPS

Check before you commit

Finding a median before ordering
Dividing a weighted total by the number of categories
Averaging group means without group sizes
Calling the largest value the mode
Ignoring an outlier
Rounding before the final step

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Mean, Median, Mode, and Weighted Mean FAQ

Can there be more than one mode?

Yes. Two values tied for highest frequency make the set bimodal.

When is median better than mean?

When the distribution is skewed or contains extreme outliers.

Do weights need to total 100%?

They must share a common total; percentages usually total 100%.

RELATED COMPETENCIES

Continue your mathematics review.

SAVE AND CONTINUE

Your progress stays on this browser.

Mastery results save to your Teacher Abi study profile.

Return to Student Hub View UPCAT Coverage

Permutations and Combinations UPCAT Reviewer: Lesson and Practice

TEACHER ABI UPCAT MATHEMATICS

Permutations and Combinations

Decide whether order changes the outcome before choosing a formula.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Order is the deciding question

Use a permutation when rearranging or assigning distinct positions. Use a combination when selecting a group whose order does not matter.

nPᵣ = n!/(n−r)!nCᵣ = n!/[r!(n−r)!]

Different roles mean order

President–secretary is different from secretary–president.

Committees ignore order

The same members form one committee regardless of listing order.

DO IT FAST

Arrange = P; choose = C

“Choose 3 from 8” gives C(8,3) = 56. “Assign three medals among 8” gives 8P3 = 336.

Why it works

A permutation counts each ordering; dividing by r! collapses those orderings into one combination.

WORKED EXAMPLES

Five forms you should recognize

1. Multiplication principle

Problem: A canteen offers 3 main dishes and 4 drinks. If a meal contains one main and one drink, how many different meals are possible?

3 × 4 = 12

Why: Each main can be paired with any of the four drinks.

2. Permutation

Problem: Seven finalists compete for president and vice president. How many officer assignments are possible?

7P2 = 7×6 = 42

Why: The positions are different, so reversing two students creates a different assignment.

3. Combination

Problem: A three-person committee is selected from eight students. How many committees are possible?

C(8,3) = 56

Why: The same three students form one committee regardless of listing order.

4. Repeated objects

Problem: How many distinct arrangements can be made from the letters of MAMA?

4!/(2!2!) = 6

Why: Divide by 2! for the repeated M’s and 2! for the repeated A’s.

5. Restriction

Problem: Five students line up, but Ana and Ben must stand together. How many arrangements are possible?

4! × 2 = 48

Why: Treat the pair as one block, then count its two internal orders: AB and BA.

COMMON TRAPS

Check before you commit

Using combinations for officer positions
Using permutations for a committee
Allowing a leading zero in a number
Forgetting repeated letters
Treating circular arrangements as linear
Counting “at least one” case by case when a complement is faster

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Permutations and Combinations FAQ

When does order matter?

When changing positions, ranks, roles, or sequence creates a different outcome.

Why is circular arrangement (n−1)!?

Rotations of the same seating are equivalent, so fix one person as a reference.

How do I handle “at least one”?

Often count all outcomes and subtract the outcomes with none.

RELATED COMPETENCIES

Continue your mathematics review.

SAVE AND CONTINUE

Your progress stays on this browser.

Mastery results save to your Teacher Abi study profile.

Return to Student Hub View UPCAT Coverage

Sets and Venn Diagrams UPCAT Reviewer: Lesson and Practice

TEACHER ABI UPCAT MATHEMATICS

Sets and Venn Diagrams

Translate the set expression or fill the smallest overlap first—then count each region exactly once.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Sets and Venn Diagrams

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UPCAT set problems test translation and overlap control

Union A∪B includes A, B, and their overlap. Intersection A∩B keeps only what the sets share. Complement A′ means outside A but still inside the universal set.

|A∪B| = |A| + |B| − |A∩B||A∪B∪C| = |A|+|B|+|C| − |A∩B|−|A∩C|−|B∩C| + |A∩B∩C|

For three sets, subtract the pairwise overlaps, then add the center back once. It was added three times with the set totals and removed three times with the pairwise subtractions.

Translate before calculating

For (A∪B)∩C′, first keep A or B, then remove every part inside C.

Start at the center

In three-set problems, fill the all-three region before converting pairwise totals into “exactly two” regions.

DO IT FAST

Center → pair-only regions → single-only regions → neither

Example: In a club survey, 15 students belong to both A and B, and 6 of those students also belong to C. The A-and-B-only region is 15−6 = 9. Repeat this subtraction for the other pairwise totals before finding the single-only regions.

Why it works

Published intersection totals normally include the center. Separating the center first prevents it from being counted repeatedly.

WORKED EXAMPLES

Five forms you should recognize

1. Two-set inclusion–exclusion

Problem: Forty-five students joined Math Club, 38 joined Science Club, and 15 joined both. How many joined at least one club?

|M∪S| = 45 + 38 − 15 = 68

Why: Adding 45 and 38 counts the 15 students in both clubs twice, so subtract the overlap once.

2. Recover an overlap

Problem: Of 68 students who joined Math or Science, 45 joined Math and 38 joined Science. How many joined both?

|M∩S| = 45 + 38 − 68 = 15

Why: The amount by which the two set totals exceed the union is the duplicated overlap.

3. Three-set union

Problem: Thirty-eight students use app A, 32 use B, and 27 use C. The pairwise intersections are 15, 12, and 11, while 6 use all three. How many use at least one app?

38 + 32 + 27 − 15 − 12 − 11 + 6 = 65

Why: Subtract the three pairwise overlaps, then restore the all-three group once.

4. Exactly two of three

Problem: The pairwise totals are 18, 15, and 16, and 7 students belong to all three sets. How many belong to exactly two?

(18−7) + (15−7) + (16−7) = 28

Why: Each pairwise total includes the same seven students in the center.

5. Minimum overlap

Problem: In a group of 100, set A has 64 members and set B has 53. What is the smallest possible overlap?

64 + 53 − 100 = 17

Why: The 117 memberships cannot fit into 100 distinct people without at least 17 belonging to both sets.

COMMON TRAPS

Check before you commit

Treating “or” as exactly one
Forgetting that pairwise totals include the triple intersection
Subtracting the all-three region instead of adding it back in the union formula
Using a set total as an “only” region
Ignoring the universal-set limit
Accepting impossible survey data

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Sets and Venn Diagrams FAQ

What does “or” mean in sets?

Inclusive or: A or B or both, unless the problem explicitly says exactly one.

Why is the triple intersection added back?

Adding the three set totals counts it three times, while subtracting the three pairwise intersections removes it three times; adding it once leaves one correct count.

How do I find a minimum possible overlap?

Add the two set totals. Any amount beyond the universal-set size must be shared.

RELATED COMPETENCIES

Continue your mathematics review.

SAVE AND CONTINUE

Your progress stays on this browser.

Mastery results save to your Teacher Abi study profile.

Return to Student Hub View UPCAT Coverage

Tuesday, July 28, 2026

Kasingkahulugan sa Konteksto UPCAT Reviewer: Lesson and Practice

TEACHER ABI UPCAT LANGUAGE PROFICIENCY

Kasingkahulugan sa Konteksto

Piliin ang salitang nagpapanatili ng tiyak na diwa, antas, tono, at gamit ng salitang nasa pangungusap.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Kasingkahulugan sa Konteksto

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Magkaugnay ang maraming salita, ngunit hindi lahat ay maaaring ipagpalit sa bawat konteksto

Unawain muna ang kahulugan ng salita sa pangungusap. Pagkatapos, subukan ang pamalit at tiyaking hindi nagbabago ang tindi, tono, o bahagi ng pananalita. Halimbawa, ang banayad ay maaaring mangahulugang hindi matindi, hindi awtomatikong mabuti.

Diwa bago salita

Bumuo muna ng payak na kahulugan bago tingnan ang mga opsiyon.

Panatilihin ang antas

Ang masidhi, malakas, at marubdob ay magkakaugnay ngunit maaaring magkaiba ang dating.

DO IT FAST

Ipaliwanag sa payak na salita, ipalit, at ihambing ang dalawang pinakamalapit

“Masusi ang pagsusuri.” Ang ibig sabihin ay sinuri ang mga detalye nang mabuti. Sa mga pagpipilian, detalyado ang nagpapanatili ng diwa; ang malawak ay maaaring maraming saklaw ngunit hindi laging masusi.

Why it works

Inaalis nito ang mga salitang kaugnay lamang ngunit hindi tunay na kapalit.

WORKED EXAMPLES

Five forms you should recognize

1. Pag-iingat

masinop → maingat

2. Tindi

marubdob → masidhi

3. Suporta

pinagtibay → sinuportahan

4. Paraan

tahas → tuwiran

5. Lalim

masusi → detalyado

COMMON TRAPS

Check before you commit

Pagpili ng pinakapamilyar
Pagkalito sa sanhi at kahulugan
Pagbabago ng tindi
Paglimot sa matalinghagang gamit
Maling bahagi ng pananalita
Pag-aakalang lahat ng kaugnay ay kasingkahulugan

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Kasingkahulugan sa Konteksto FAQ

Maaari bang dalawang sagot ang magkaugnay?

Oo, ngunit isa lamang ang dapat magpanatili ng eksaktong diwa at tono sa pangungusap.

Kailangan bang kabisaduhin ang lahat ng malalalim na salita?

Makatutulong ang bokabularyo, ngunit mahalaga rin ang context clues at pagsusubok ng pamalit.

RELATED COMPETENCIES

Continue your language review.

SAVE AND CONTINUE

Your progress stays on this browser.

Mastery results save to your Teacher Abi study profile.

Return to Student Hub View UPCAT Coverage