【People's Education Press】Junior High School Math, Grade 7, Part B: The Encounter of Angles – From Vertical Angles to the Special Case of Perpendicularity

Imagine a pair of scissors fully opened or the starting line of a school track. When the two blades meet, the magic of geometry begins. At this point of intersection, angles appear in pairs—some complement each other to form a straight angle of 180°, others mirror each other across the vertex. When these two lines reach their most 'rigid' configuration—when one of the angles becomes 90°—they enter intoperpendicularitythis ultimate and unique state of balance.

Fundamental Relationships in Intersecting Lines

In the same plane, when two lines intersect, two important angular relationships emerge:

Adjacent Supplementary Angles (Angles on a Straight Line): They share a common side $OC$, and their other sides are opposite extensions of each other. Numerically, adjacent supplementary angles are supplementary (their sum is $180^\circ$).
Vertical Angles (Opposite Angles): They share a common vertex $O$, and the sides of one angle are the opposite extensions of the sides of the other angle.

Deductive Reasoning: Vertical Angles Are Equal

Why are vertical angles always equal? Let’s deconstruct this with rigorous logic:

$because$ $\angle 1$ and $\angle 2$ are supplementary (definition of adjacent supplementary angles)

$because$ $\angle 3$ and $\angle 2$ are supplementary (definition of adjacent supplementary angles)

$\therefore$ $\angle 1 = \angle 3$ (Supplements of the Same Angle Are Equal)

Perpendicularity: A Special Position of Intersection

Perpendicular (Perpendicular) is an extreme case of intersection. When two lines intersect and one of the four resulting angles is $90^\circ$, the two lines are perpendicular to each other. One line is called theperpendicular line, and their point of intersection is called thefoot of the perpendicular.

Core Criteria and Properties

Symbolic Notation: If lines $a$ and $b$ are perpendicular, write $a \perp b$; if segments $AB$ and $CD$ are perpendicular, write $AB \perp CD$.
Perpendicular Postulate: In the same plane, through a given point, there exists exactly one line perpendicular to a given line. This establishes theuniqueness.
Shortest Perpendicular Segment: Among all segments connecting a point outside a line to points on the line, the perpendicular segment is the shortest.

🎯 Core Principle

从“相交”到“垂直”，是角度从变动到定格的过程。掌握符号 $ecause$ (因为) 与 $ herefore$ (所以) 的规范表述，是跨入几何证明大门的钥匙。

$\angle AOC = 90^\circ \iff AB \perp CD$

QUESTION 1

As shown in the figure, lines $a$ and $b$ intersect, and it is given that $\angle 1 = 40^\circ$. What are the measures of $\angle 2$ and $\angle 3$?

$\angle 2 = 140^\circ, \angle 3 = 40^\circ$

$\angle 2 = 40^\circ, \angle 3 = 140^\circ$

$\angle 2 = 140^\circ, \angle 3 = 140^\circ$

$\angle 2 = 50^\circ, \angle 3 = 40^\circ$

QUESTION 2

When the four angles formed by the intersection of two lines are all equal, what is their positional relationship?

Parallel

Mutually Perpendicular

Coincident

Cannot be Determined

QUESTION 3

Which of the following statements about perpendicular lines is correct?

Infinitely many perpendicular lines can be drawn through a point to a given line

In the same plane, through a point, there is exactly one line perpendicular to a given line

The perpendicular segment is the distance from a point to a line

Two lines intersecting must be perpendicular

QUESTION 4

As shown in the figure, take two wooden strips nailed together to form an intersecting model. If $\angle \alpha = m^\circ$, what is the measure of the adjacent angle?

$m^\circ$

$(180 - m)^\circ$

$(90 - m)^\circ$

Cannot be calculated

QUESTION 5

When laying railway tracks, given that the angle between the rail and the sleeper is $\angle 2 = 90^\circ$, which angle should be measured to determine if the two rails are parallel?

Measure $\angle 5$; if $\angle 5 = 90^\circ$, then they are parallel

Measure $\angle 5$; if $\angle 5 = 45^\circ$, then they are parallel

Just look at $\angle 2$

Measure the length of the sleeper

QUESTION 6

As shown in the figure, $PO \perp l$, and point A moves along line $l$. Compare the lengths of segments $PO$ and $PA$. What is the conclusion?

$PO > PA$

$PO = PA$

$PO \le PA$

Cannot be compared

QUESTION 7

What do the symbols '$\because$' and '$\therefore$' represent, respectively?

Because, Therefore

Therefore, Because

Equal, Congruent

Foot of the perpendicular, Perpendicular

QUESTION 8

If $\angle 1$ and $\angle 2$ are adjacent supplementary angles, and $\angle 1 = 35^\circ$, what is the measure of $\angle 2$?

$35^\circ$

$145^\circ$

$55^\circ$

$155^\circ$

QUESTION 9

What is the essence of drawing a perpendicular to a line segment?

Drawing a perpendicular through the midpoint of the segment

Drawing a perpendicular to the line containing the segment

Drawing a perpendicular at the endpoint of the segment

Drawing a longer line segment