People's Education Press: Middle School Math, Grade 8, Part 1: Overlapping Figures and Properties: Exploring Congruent Triangles

Observe the steel truss structure of Nanjing Yangtze River Bridge, and you'll notice countless triangular units connected together. Because these trianglesare congruentandhave equal corresponding sides, they maintain extreme geometric stability under external forces. This property of 'perfect overlap' is not only the foundation of engineering but also the essence of geometric logic.

The Essence of Congruent Triangles: The Property of Overlap

When we move a figure through translation, reflection, or rotation so that it perfectly overlaps with another, we have successfully transformed the concept of 'congruence' from real-world objects to geometric models.

Congruent Figures (Congruent figures): Two figures that can be perfectly overlapped.
Congruent Triangles (Congruent triangles): Two triangles that can be perfectly overlapped.

During the overlapping process,the overlapping verticesare called corresponding vertices,the overlapping sidesare called corresponding sides,the overlapping anglesare called corresponding angles.

Symbol and Expression

Congruence is represented by the symbol "$\cong$", read as "congruent to".

Note: When writing two triangles as congruent, it is usually standard to place the letters representingcorresponding vertices in corresponding positions. For example, $\triangle ABC \cong \triangle DBC$ means $A$ corresponds to $D$, $B$ to $B$, and $C$ to $C$.

Core Property

In congruent triangles,have equal corresponding sides, and in congruent triangles,corresponding angles are equal.

🎯 Identification Tips

In complex figures, pay attention to identifying 'common sides' (e.g., $AD$, which is a side of both $\triangle ABD$ and $\triangle ACD$) or 'common angles'. These are key clues for determining congruence correspondence.

QUESTION 1

What principle must be followed when writing the congruence relationship $\triangle ABC \cong \triangle DEF$?

Letters must be arranged in alphabetical order

The letters representing corresponding vertices must be written in corresponding positions

Write the larger triangle first

There are no special requirements; write freely

QUESTION 2

As shown in Figure 12.1-2(2), $\triangle ABC$ and $\triangle DBC$ are symmetric about line $BC$. Which statement about corresponding elements is incorrect?

$AB$ and $DB$ are corresponding sides

$BC$ is a common side and also a corresponding side

$\angle A$ and $\angle D$ are corresponding angles

$AC$ and $BC$ are corresponding sides

QUESTION 3

Given $\triangle OCA \cong \triangle OBD$, where point $C$ corresponds to point $B$, and point $A$ to point $D$. Which conclusion is correct?

$OC = OB, OA = OD, AC = BD

$OC = OD, OA = OB, AC = BD

$OA = AC, OB = BD

$\angle COA = \angle DOB = 90^\circ$

QUESTION 4

As shown in the figure, $\triangle ABC \cong \triangle CDA$, with $AB$ and $CD$, $BC$ and $DA$ being corresponding sides. What is the corresponding angle to $\angle BAC$?

$\angle CAD$

$\angle DCA$

$\angle D$

$\angle B$

QUESTION 5

The figure shows two congruent triangles. The first triangle has angles of $54^\circ$ and $60^\circ$, and sides $a, b, c$. In the second triangle, sides $b$ and $c$ are known. What is the measure of the angle $\angle 1$ between them?

$54^\circ$

$60^\circ$

$66^\circ$

$114^\circ$

QUESTION 6

$\triangle EFG \cong \triangle NMH$, with $\angle F = \angle M$. Given $EF = 2.1$ cm, $EH = 1.1$ cm, $NH = 3.3$ cm. Find the length of segment $NM$.

$1.1$ cm

$2.1$ cm

$3.3$ cm

$2.2$ cm

QUESTION 7

Following the previous question, find the length of segment $HG$.

$1.1$ cm

$2.2$ cm

$3.3$ cm

$4.4$ cm

QUESTION 8

$\triangle ABC \cong \triangle DEC$, with $CA$ and $CD$, $CB$ and $CE$ being corresponding sides. What is the relationship between $\angle ACD$ and $\angle BCE$?

complementary

equal

$\angle ACD = 2\angle BCE$

no necessary relationship

QUESTION 9

$\triangle AEC \cong \triangle ADB$, with points $E$ and $D$ as corresponding vertices. If $\angle A = 50^\circ$, $\angle ABD = 39^\circ$, find the measure of $\angle AEC$.

$91^\circ$

$39^\circ$

$50^\circ$

$100^\circ$

QUESTION 10

"Congruence" is everywhere in life. Which of the following pairs of objects best fits the definition of "congruent figures"?

Two identical-value stamps printed from the same layout

One large cat and one small cat

Two drawings from a copier with different scales

Two circles drawn with a compass of different radii