Angle in degrees between the hands of a watch; congruence of triangles

Expert intro: This set combines angle-measure problems with angle-bisector proofs and the definition of triangle congruence. The main skill is to turn each statement into a precise angle relation or a side-angle-side comparison.

Detailed walkthrough

What this exercise set is testing

The questions in this excerpt focus on two linked ideas:

1. Angle measurement and rotation — especially with clock hands and angles turned in a given time.
2. Angle bisectors and triangle congruence — especially proofs using symmetry, adjacent angles, and bisectors.

Core methods you should use

1) Clock-hands angle

For a clock, use:

• minute hand speed = 6° per minute
• hour hand speed = 0.5° per minute
• second hand speed = 6° per second

So for a time like 5:45, first find the positions of both hands and then take the smaller angle between them.

2) Angles turned in 20 minutes

Multiply the angular speed by time:

• minute hand: $6\times 20=120^\circ$
• hour hand: $0.5\times 20=10^\circ$
• second hand in 20 minutes: $6\times 60\times 20=7200^\circ$

If the question asks for a full number of turns, convert degrees to revolutions using $360^\circ$ per turn.

3) Midpoint and extension identities

If $C$ is the midpoint of $AB$ and $AB$ is produced to $D$, then place the points on a line and write distances algebraically.

A clean coordinate choice is:

• let $C=0$
• let $A=-a$
• let $B=a$
• let $D=d$ with $d>a$

Then

$AD=d+a,\quad BD=d-a,\quad CD=d,$

so

$AD+BD=(d+a)+(d-a)=2d=2CD.$

The same idea works when $D$ is any point on $CB$.

4) Angle bisector relations

If $OC$ bisects $\angle AOB$, then

$\angle AOC=\angle COB=\tfrac12\angle AOB.$

This is the key fact used in statements like

$\angle XOA+\angle XOB=2\angle XOC.$

The proof usually comes from splitting the larger angle into two equal halves and comparing the sum of the adjacent angles.

5) Perpendicular bisectors of adjacent angles

If the bisectors of two adjacent angles are perpendicular, then the original angles together must form a straight line. This is because each bisector contributes half of its angle, and the right angle between bisectors forces the two original angles to sum to $180^\circ$.

Triangle congruence definition

The excerpt then begins congruence of triangles.

Two triangles are congruent if:

• two sides of one are equal to two sides of the other, and
• the included angles between those sides are equal.

This is the SAS congruence criterion.

So if

$A_1B_1=A_2B_2,\quad B_1C_1=B_2C_2,\quad \angle B_1=\angle B_2,$

then

$\triangle A_1B_1C_1\cong\triangle A_2B_2C_2.$

How to read the rest of the problems

For each proof-style question in the list:

• identify the midpoint, bisector, or adjacency condition;
• write the angle or distance equality explicitly;
• use symmetry or decomposition of angles;
• conclude the required equality step by step.

That pattern is enough for all the numbered statements in this excerpt.

💡 Pitfall guide

A frequent mistake is to mix up the larger angle and the smaller angle when working with clock hands or intersecting lines. Another common error is to forget that an angle bisector creates two equal angles, not two equal sides. For congruence questions, do not claim triangle congruence unless the matching sides and included angle are clearly identified.

🔄 Real-world variant

If the time in the clock problem changes, the method stays the same: compute each hand’s angular displacement from 12 o’clock, then find the difference. If the geometry statements change from adjacent angles to vertically opposite angles or exterior angles, the same strategy still applies: rewrite everything in terms of equal angle halves and line sums of $180^\circ$.

🔍 Related terms

clock angle, angle bisector, SAS congruence