This MCQ module is based on: Angle Sum Property and Altitudes
TOPIC 26 OF 31
Angle Sum Property and Altitudes
🎓 Class 7
Mathematics
CBSE
Theory
Ch 7 — A Tale of Three Intersecting Lines
⏱ ~35 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Angle Sum Property and Altitudes
Targeting Class 7 level in Geometry, with Basic difficulty.
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7.3 Constructing Triangles When Some Sides & Angles are Given
Two Sides and the Included Angle (SAS)
If two sides and the angle between them are given, the triangle is completely determined.
Example. Construct \(\triangle ABC\) with AB = 5 cm, AC = 4 cm and \(\angle A = 45^\circ\).
Step 1. Draw AB = 5 cm.
Step 2. At A, draw a ray making \(45^\circ\) with AB.
Step 3. On this ray, mark C so that AC = 4 cm.
Step 4. Join BC.
Fig 7.6 — SAS construction (two sides and the included angle).
Two Angles and the Included Side (ASA)
If the side between two known angles is given, the triangle is also uniquely determined.
Example. \(AB = 5\) cm, \(\angle A = 45^\circ\), \(\angle B = 80^\circ\). Draw AB, then rays at A and B of the given measures; their intersection is C.
For such a triangle to exist, the two given angles must satisfy \(0^\circ < \angle A + \angle B < 180^\circ\). If they add to \(180^\circ\) or more, the two rays are parallel or diverge, and no triangle forms.
Angle Sum Property
What is the sum of the three angles of any triangle? We draw a line XY through vertex A parallel to BC.
Fig 7.7 — XY ∥ BC through A. Alternate angles: ∠XAB = ∠B, ∠YAC = ∠C.
Since XY is a straight line through A, \(\angle XAB + \angle BAC + \angle YAC = 180^\circ\). By the alternate-angle property for transversals, \(\angle XAB = \angle B\) and \(\angle YAC = \angle C\). Therefore:
\(\angle A + \angle B + \angle C = 180^\circ\)
Angle Sum Property
In every triangle the sum of the three interior angles equals \(180^\circ\).Historical note
This result first appears in Euclid's Elements (≈ 300 BCE). It is one of the earliest theorems rigorously proved in mathematics.Exterior Angle of a Triangle
If a side of the triangle is extended, the angle formed outside the triangle with the adjacent side is called an exterior angle?.
Fig 7.8 — ∠ACD is an exterior angle of \(\triangle ABC\).
Since \(\angle ACB + \angle ACD = 180^\circ\) (linear pair) and \(\angle A + \angle B + \angle ACB = 180^\circ\), we conclude: exterior angle = sum of the two opposite interior angles, i.e. \(\angle ACD = \angle A + \angle B\).
Worked example. If \(\angle A = 50^\circ\) and \(\angle B = 60^\circ\), then \(\angle ACB = 180^\circ - 50^\circ - 60^\circ = 70^\circ\); exterior \(\angle ACD = 180^\circ - 70^\circ = 110^\circ\), which equals \(50^\circ + 60^\circ\). ✓
7.4 Altitudes of a Triangle
An altitude? is the perpendicular from a vertex to the line containing the opposite side. The foot of the perpendicular is where the altitude meets that line. Every triangle has three altitudes — one from each vertex.
Fig 7.9 — Altitude AD from A, perpendicular to BC.
For a right-angled triangle, two of the three altitudes coincide with the two legs (the sides making the right angle). For an obtuse triangle, two altitudes fall outside the triangle — the opposite side must be extended before the perpendicular foot can be marked.
Altitudes Using Paper Folding
Cut a paper triangle. To drop the altitude from A to BC, fold the paper so that BC folds onto itself and the crease passes through A. The crease is the altitude.
Figure it Out
Q1. Find the third angle of \(\triangle ABC\) if (a) \(\angle A=36^\circ,\angle B=72^\circ\), (b) \(\angle A=150^\circ,\angle B=15^\circ\), (c) \(\angle A=90^\circ,\angle B=30^\circ\), (d) \(\angle A=75^\circ,\angle B=45^\circ\).
Use \(\angle A+\angle B+\angle C=180^\circ\). (a) 72°; (b) 15°; (c) 60°; (d) 60°.
Q2. Can you construct a triangle all of whose angles equal 70°? If two angles are 70°, what is the third? If all angles are equal, what must each be?
Three 70° angles sum to 210° ≠ 180°, so no. If two are 70° each, the third is \(180-140=40^\circ\). If all three are equal, each is \(180/3=60^\circ\) (equilateral triangle).
Q3. In \(\triangle ABC\), \(\angle B = \angle C\) and \(\angle A = 50^\circ\). Find \(\angle B\) and \(\angle C\).
\(\angle B + \angle C = 130^\circ\) and they are equal, so \(\angle B = \angle C = 65^\circ\).
Q4. Determine whether each pair can be the two known angles of a triangle: (a) 35°, 150° (b) 70°, 30° (c) 90°, 85° (d) 50°, 150°.
Need sum < 180°: (a) 185° ❌; (b) 100° ✅; (c) 175° ✅; (d) 200° ❌.
Activity: Tearing the Corners
Materials: Paper, scissors.
- Cut any triangle ABC from paper.
- Tear off each of the three corners (with a bit of the two sides attached).
- Place the three corners next to each other on a straight line, sharing a common vertex.
- Observe how they arrange.
The three corners together form a straight angle (180°), giving a visual, no-words proof that \(\angle A+\angle B+\angle C=180^\circ\).
Competency-Based Questions
Scenario: An architect is designing a triangular roof truss \(\triangle PQR\) over a hall. Two of the angles are \(\angle P = 40^\circ\) and \(\angle Q = 70^\circ\).
Q1. Find the third angle \(\angle R\).
L3 Apply\(\angle R = 180^\circ - 40^\circ - 70^\circ = \mathbf{70^\circ}\).
Q2. The architect observes \(\angle Q = \angle R\). What kind of triangle is PQR?
L4 AnalyseTwo equal angles imply the sides opposite them are equal, so PQR is isosceles with PQ = PR.
Q3. The client suggests making \(\angle P = 95^\circ,\ \angle Q = 50^\circ,\ \angle R = 40^\circ\). Evaluate whether this can be built.
L5 EvaluateSum = 185° > 180°. This violates the angle sum property; the truss cannot be constructed.
Q4. Create a triangle specification (three angles) where the exterior angle at R equals the sum of the other two interior angles and is exactly 120°.
L6 CreateExterior at R = 120° ⇒ \(\angle P + \angle Q = 120^\circ\) and \(\angle R = 60^\circ\). Many valid answers: e.g. \(\angle P = 50^\circ,\ \angle Q = 70^\circ,\ \angle R = 60^\circ\).
Assertion–Reason Questions
A: No triangle can have two right angles.
R: The sum of the interior angles of a triangle is 180°.
R: The sum of the interior angles of a triangle is 180°.
(a) — two right angles already contribute 180°, leaving 0° for the third, impossible. R explains A.
A: An exterior angle of a triangle is always greater than each interior opposite angle.
R: Exterior angle = sum of the two opposite interior angles.
R: Exterior angle = sum of the two opposite interior angles.
(a) — since it equals their sum, it must exceed each one (all angles positive). R explains A.
A: In a right-angled triangle, two altitudes are the legs themselves.
R: An altitude is perpendicular to the opposite side, and the two legs of a right triangle are perpendicular to each other.
R: An altitude is perpendicular to the opposite side, and the two legs of a right triangle are perpendicular to each other.
(a) — each leg is perpendicular to the opposite (other) leg, so each leg itself is the altitude from its endpoint.
Frequently Asked Questions
Why do the angles of a triangle always sum to 180 degrees?
Drawing a line through one vertex parallel to the opposite side shows that the three interior angles correspond to the three parts of a straight angle, which is 180 degrees.
What is an altitude of a triangle?
An altitude is the perpendicular segment from a vertex to the line containing the opposite side. Every triangle has three altitudes, one from each vertex.
What is the exterior angle theorem?
An exterior angle of a triangle equals the sum of the two opposite (non-adjacent) interior angles. For example, if interior angles are 50 and 60 degrees, the exterior at the third vertex is 110 degrees.
Where do the altitudes meet?
The three altitudes of any triangle meet at a single point called the orthocentre. It lies inside for acute triangles, on a vertex for right triangles, and outside for obtuse triangles.
How is the angle sum property used to find an unknown angle?
Subtract the sum of the two known angles from 180 degrees. If two angles are 70 and 40 degrees, the third must be 180 minus 110, which is 70 degrees.
AI Tutor
Mathematics Class 7 — Ganita Prakash
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